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THE BINARY STARS 

BY ROBERT G. AITKEN 



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{Frontispiece^ 
Plate I. — The 36-inch refractor of the Lick Observatory, 



THE BINARY STAES 



BY 

ROBERT GRANT AITKEN 

Late Director and Astronomer, Lick Observatory, 
University of California 



DOVER PUBLICATIONS, INC. 

NEW YORK 



Copyright ©1935 by the McGraw-Hill Book Company, 
Inc. 

Copyright renewed 1963 by Malcolm D. Aitken 

Copyright ©1964 by Dover Publications, Inc. 

All rights reserved under Pan American and 
International Copyright Conventions 

Published simultaneously in Canada by 
McClelland and Stewart, Limited 

Published in the United Kingdom by 
Constable and Company, Limited, 
10 Orange Street, London W.C.2 



This Dover edition, first published in 1964, is an 

unabridged and corrected republication of the second 

edition, published by the McGraw-Hill Book Company 

in 1935 



The publishers wish to thank Jack T. Kent, Associate 
Professor of Mathematics and Astronomy at Texas 
A & M University, for preparing the corrections and 
supplementary reference materials that have been 
incorporated into this Dover edition. 



Library of Congress Catalog Card Number: 64-13456 



Manufactured in the United States of America 

Dover Publications, Inc. 

180 Varick Street 

New York 14, N.Y. 



PREFACE TO THE DOVER EDITION 

During the past several years it has become more and more 
apparent that a need exists for the publication of a new edition 
of-Aitken's The Binary Stars. Since the book has been out of 
print for many years, copies of it are practically unattainable. 
With the advent of the Space Age, many of the methods presented 
in this book are applicable to other fields than binary stars, and many 
of the binary-star astronomers would either like to replace their old 
worn-out copies, or own one for the first time. It is with the idea 
of serving these people, as well as libraries, computing centers, and 
industry, that this edition is presented. 

No attempt has been made to bring the material up to date. This 
has been adequately done elsewhere, as is indicated throughout this 
book by new references. The book is simply presented as a classic 
in the field. We have attempted to eliminate all known errors, and 
to present sufficient additional footnotes and references to start the 
reader on the right path should he wish to continue his reading and 
study. 

We wish to express our appreciation to the following people for 
their suggestions and invaluable assistance in listing all known errors, 
and eliminating them: Dr. George Van Biesbroeck, Yerkes Observa- 
tory; Dr. Hamilton M. Jeffers, Lick Observatory; Dr. W. H. van 
den Bos, Union Observatory, Johannesburg, South Africa; Dr. K. A. 
Strand, U.S. Naval Observatory; and others. 

Jack T. Kent 
Texas A & M University 

November, 1963 



PREFACE 

The first edition of this book was prepared as a contribution 
to the Series of Semi-Centennial Publications issued in 1918 by 
the University of California and was included in that series, 
although published commercially. It has long been out of 
print and is now also, in large part, out of date, as a result 
of the great amount of work that has been done in the field of 
binary star astronomy in the past sixteen years. 

In its chapter headings and in the general form of presentation 
the present edition follows the plan adopted in the original work, 
but it has been necessary to revise all of the chapters and to 
rewrite some of them in large part to take account of the work 
done in recent years. One measure of this later work is given 
by the tables of orbits in the Appendix. In the first edition 
87 orbits of visual binaries and 137 orbits of spectroscopic binaries 
were listed. In the present edition, the two tables which are 
based upon all data available to me before September 1933, 
contain 116 and 326 pairs, respectively, though Cepheids and 
pseudo-Cepheids are excluded. 

It is a pleasure to express my gratitude to my colleague, 
Dr. J. H. Moore, for his kindness in revising the excellent chapter 
(V) on The Radial Velocity of a Star, which he prepared for the 
first edition; to Profs. H. N. Russell and R. S. Dugan for criticisms 
and suggestions relating to the chapter on Eclipsing Binary Stars, 
and to Dr. W. H. Van den Bos for placing data relating to the 
Thiele-Innes method at my disposal, and to acknowledge my 
indebtedness to other friends. I desire also to express again 
my thanks to all those who gave generous assistance in the 
preparation of the first edition. 

Robert Grant Aitken. 

University op California, 
April, 1935. 



CONTENTS 

Paob 

Preface vi 

Introduction ix 

CHAPTER I 
Historical Sketch: The Early Period 1 

CHAPTER II 
Historical Sketch: The Modern Period 20 

CHAPTER III 
Observing Methods, Visual Binary Stars 41 

CHAPTER IV 
The Orbit of a Visual Binary Star 70 

CHAPTER V 
The Radial Velocity of a Star, by Dr. J. H. Moore 125 

CHAPTER VI 
The Orbit of a Spectroscopic Binary Star 151 

CHAPTER VII 
Eclipsing Binary Stars 181 

CHAPTER VIII 
The Known Orbits of the Binary Stars 203 

CHAPTER IX 

Some Binary Systems of Special Interest 235 

CHAPTER X 
Statistical Data Relating to the Visual Binary 
Stars in the Northern Sky 257 

CHAPTER XI 
The Origin of the Binary Stars 273 

Appendix 283 

Table of Orbits of Visual Binary Stars 284 

Table of Orbits of Spectroscopic Binary Stars . 288 

Index 303 

vii 



INTRODUCTION 

It is the object of this volume to give a general account of 
our present knowledge of the binary stars, including such an 
exposition of the best observing methods and of approved 
methods of orbit computation as may make it a useful guide 
for those who wish to undertake the investigation of these 
systems; and to present some conclusions based upon the author's 
own researches during the past forty years. 

The term binary star was first used by Sir William Herschel, 
in 1802, in his paper "On the Construction of the Universe," 
to designate "a real double star — the union of two stars, that 
are formed together in one system, by the laws of attraction." 

The term double star is of earlier origin; its Greek equivalent 
was, in fact, used by Ptolemy to describe the appearance of 
v Sagittarii, two fifth-magnitude stars whose angular separation 
is about 14/, or a little less than half of the Moon's apparent 
diameter. It is still occasionally applied to this and other 
pairs of stars visible to the unaided eye, but is generally used 
to designate pairs separated by not more than a few seconds 
of arc and therefore visible as two stars only with the aid of a 
telescope. 

Not every double star is a binary system, for, since all of the 
stars are apparently mere points of light projected upon the 
surface of the celestial sphere, two unrelated stars may appear 
to be closely associated simply as the result of the laws of per- 
spective. Herschel draws the distinction between the two 
classes of objects in the following words: 

... if a certain star should be situated at any, perhaps immense, 
distance behind another, and but little deviating from the line in which 
we see the first, we should have the appearance of a double star. But 
these stars being totally unconnected would not form a binary system. 
If, on the contrary, two stars should really be situated very near each 
other, and at the same time so far insulated as not to be materially 
affected by neighboring stars, they will then compose a separate system, 
and remain united by the bond of their mutual gravitation toward each 
other. This should be called a real double star. 



X INTRODUCTION 

Within the last half century we have become acquainted 
with a class of binary systems which are not double stars at 
all in the ordinary sense of the term, for the two component 
stars are not separately visible in any telescope. These are 
the spectroscopic binary stars, so named because their existence 
is demonstrated by a slight periodic shifting to and fro of the 
lines in their spectra, which, as will be shown, is evidence of a 
periodic variation in the radial velocity (the velocity in the 
line of sight, toward or away from the observer) of the star. 
With the possible exception of factors which may be introduced 
by the fact that the distance between the two components of a 
spectroscopic binary is, in general, so small (tidal interactions, 
for example), there seem to be no dynamical differences between 
the spectroscopic and the visual binary systems. The two classes 
will, therefore, be regarded in this volume as members of a 
single species. 



THE BINARY STARS* 

CHAPTER I 
HISTORICAL SKETCH: THE EARLY PERIOD 

The first double star was discovered about the year 1650 
by the Italian astronomer, Jean Baptiste Riccioli. This was 
f Ursae Majoris (Mizar). It is a remarkable coincidence that 
Mizar was also the first double star to be observed photographi- 
cally, measurable images being secured by G. P. Bond, at the 
Harvard College Observatory in 1857; and that its principal 
component was the first spectroscopic binary to be discovered, 
the announcement being made by E. C. Pickering in 1889. 

In 1656, Huygens saw 6 Ononis resolved into the three prin- 
cipal stars of the group which form the familiar Trapezium, 
and, in 1664, Hooke noted that y Arietis consisted of two stars. 
At least two additional pairs, one of which proved to be of 
more than ordinary interest to astronomers, were discovered 
before the close of the seventeenth century. It is worthy 
of passing note that these were southern stars, not visible from 
European latitudes — a Cruris, discovered by the Jesuit mis- 
sionary, Father Fontenay, at the Cape of Good Hope, in 
1685, and a Centauri, discovered by his confrere, Father Richaud, 
while observing a comet at Pondicherry, India, in December, 
1689. 

These discoveries were all accidental, made in the course of 
observations taken for other purposes. This is true also of the 
double stars found in the first three-quarters of the eighteenth 
century. Among these were the discoveries of y Virginis, in 
1718, and of Castor, in 1719, by Bradley and Pound, and of 
61 Cygni, by Bradley, in 1753. 

No suspicion seems to have been entertained by these astron- 
omers or by their contemporaries that the juxtaposition of the 
two star images in such pairs was other than optical, due to 
the chance positions of the Earth and the two stars in nearly 

* [See G. Van Biesbroeck, in the book review, Ap. Jour. 82, 368, 1935. — 
J.T.K.] 

1 



2 THE BINARY STARS 

a straight line. They were therefore regarded as mere curiosi- 
ties, and no effort was made to increase their number; nor 
were observations of the relative positions of the two com- 
ponents recorded except in descriptive terms. Father Feuille, 
for instance, on July 4, 1709, noted that the fainter star in the 
double, a Centauri, "is the more western and their distance is 
equal to the diameter of this star," and Bradley and Pound 
entered in their observing book, on March 30, 1719, that "the 
direction of the double star a of Gemini was so nearly parallel 
to a line through k and <r of Gemini that, after many trials, we 
could scarce determine on which side of <r the line from *c par- 
allel to the line of their direction tended; if on either, it was 
towards /3." 

Halley's discovery, in 1718, that some of the brighter stars, 
Sirius, Arcturus, Aldebaran, were in motion, having unmis- 
takably changed their positions in the sky since the time of 
Ptolemy, unquestionably stimulated the interest of astron- 
omers in precise observations of the stars. These researches 
and their results, in turn, were probably largely responsible for 
the philosophical speculations which began to appear shortly 
after the middle of the eighteenth century as to the possi- 
bility of the existence of systems among the stars. Famous 
among the latter are the Cosmologische Brief e* published in 
1761 by Lambert, in which it is maintained that the stars are 
suns and are accompanied by retinues of planets. Lambert, 
however, apparently did not connect his speculations with the 
double stars then known. Six years later, in 1767, John Michell, 
in a paper read before the Royal Society of London, presented 
a strong argument, based upon the theory of probabilities, 
that "such double stars, etc., as appear to consist of two or 
more stars placed near together, do really consist of stars placed 
near together, and under the influence of some general law, 
whenever the probability is very great, that there would not 
have been any such stars so near together, if all those that are 
not less bright than themselves had been scattered at random 
through the whole heavens." Michell thus has the credit of 
being the first to establish the probability of the existence of 
physical systems among the stars; but there were no observational 

* Cosmologische Brief e uber die Einrichtung des WeUbaues, Ausgef ertigt 
von J. H. Lambert, Augsburg, 1761. 



HISTORICAL SKETCH: THE EARLY PERIOD 



data to support his deductions and they had no direct influence 
upon the progress of astronomy. 

The real beginning of double star astronomy dates from the 
activities of Christian Mayer and of Sir William Herschel, 
in the last quarter of the eighteenth century. If a definite 
date is desired we may well follow Lewis in adopting the year 
1779, for that year is marked by the appearance of Mayer's 
small book entitled De novis in Coelo Sidereo Phaenominis in 
miris Stellarum fixarum Comitibus, wherein he speculates upon 
the possibility of small suns revolving around larger ones, and 
by the beginning of Herschel's systematic search for double 
stars. 

The difference between Mayer's speculations and earlier ones 
is that his rest in some degree at least upon observations. These 
were made with an 8-ft. Bird mural quadrant at Mannheim, 
in 1777 and 1778. At any rate, in his book just referred to, 
he publishes a long list of faint companions observed in the 
neighborhood of brighter stars.* As one result of his observa- 
tions he sent to Bode, at Berlin, the first collection or catalogue 
of double stars ever published. The list contained earlier dis- 
coveries as well as his own and is printed in the Astronomisches 
Jahrbuch for the year 1784 (issued in 1781) under the caption 
"Verzeichnis aller bisher entdeckten Doppelsterne." The fol- 
lowing tabulation gives the first five entries: 









Unterschied 








Gerade 
Aufst. 


Abwei- 
chung 






Stellung 


Grdsse 


in der 


in der 


Abstand 


des 
Klei- 








Aufst. 


Abw. 






G. M. 


G. M. 


Sec. 


Sec. 


Sec. 


nern 


Andromeda beyde 9ter 


8 38 


29 45N 


45 


24 


46 


S. W. 


Andromeda beyde 9ter 


13 13 


2018N 


15 


29 


32 


S. O. 


f Fische 6. und 7ter 


15 33 


6 25N 


22 


9 


24 


N. 0. 


bey n Fische beyde 7ter 


19 24 


5 ON 





4 


4 


s. 


y Widder beyde 5ter 


25 22 


1813N 


3 


12 


12 


s. w. 



In all, there are 80 entries, many of which, like Castor and 
y Virginis, are among the best known double stars. Others 

* This list, rearranged according to constellations, was reprinted by 
Schjellerup in the journal Copernicus, 3, 57, 1884. 



4 THE BfNAHY STARS 

are too wide to be found even in Herschel's catalogues and a 
few cannot be identified with certainty. Southern pairs, like 
a Ceniauri, are of course not included, and, curiously enough, 
6 Orionis is not listed. The relative positions given for the 
stars in each pair are little better than estimates, for precise 
measures were not practicable until the invention of the "revolv- 
ing micrometer." 

In his comments on Mayer's catalogue, Bode points out that 
careful observations of such pairs might become of special 




N 
\ 



Sir William Ilerachel. 

value in the course of time for the discovery of proper motions, 
since it would be possible to recognize the fact of motion in 
one or the other star as soon as the distance between them had 
changed by a very few seconds of arc. Mayer himself seems 
to have had proper motions in view in making his observations 
and catalogue rather than any idea of orbital motions. 

Sir William Herschel "began to look at the planets and the 
stars" in May, 1773; on Mar. 1, 1774, "he commenced his 
astronomical journal by noting that he had viewed Saturn's 
ring with a power of forty, appearing 'like two slender arms' 
and also 'the lucid spot in Orion's sword belt.'" The earliest 
double star measure recorded in his first catalogue is that of 
& Orionis, on Nov. 11, 1776, and he made a few others in the 



HISTORICAL SKETCH: THE EARLY PERIOD 5 

two years following. It was not until 1779, however, that he 
set to work in earnest to search for these objects, for it was 
then that he conceived the idea of utilizing them to test a method 
of measuring stellar parallax suggested long before by Galileo. 
The principle involved is very simple. If two stars are in 
the same general direction from us and one is comparatively 
near us while the other is extremely distant, the annual revolu- 
tion of the Earth about the Sun will produce a periodic variation 
in the relative positions of the two. As a first approxima- 
tion, we may regard the more distant star as absolutely fixed 
and derive the parallax of the nearer one from the measured 
displacements. 

It seemed clear to Herschel that the objects best fitted for 
such an investigation were close double stars with components 
of unequal brightness. He pointed out in his paper On the 
Parallaxes of the Fixed Stars, read before the Royal Society 
in 1781, that the displacement could be more easily and cer- 
tainly detected in a close double star than in a pair of stars 
more widely separated and also that in the former case the 
observations would be free from many errors necessarily affecting 
the measures in the latter. 

As soon as I was fully satisfied [he continues] that in the investigation 
of parallax the method of double stars would have many advantages 
above any other, it became necessary to look out for proper stars. This 
introduced a new series of observations. I resolved to examine every 
star in the heavens with the utmost attention and a very high power, 
that I might collect such materials for this research as would enable 
me to fix my observations upon those that would best answer my ends. 

In this reasoning, Herschel assumes that there is no physical 
connection between the components of such close double stars, 
— a fact upon which every writer on the history of double-star 
astronomy has commented. This was not an oversight on his 
part, for at the close of his first catalogue of double stars he 
remarks : 

I preferred that expression [i.e., double stars] to any other, such as 
Comes, Companion, or Satellite; because, in my opinion, it is much too 
soon to form any theories about small stars revolving round large ones, 
and I therefore thought it advisable carefully to avoid any expression 
that might convey that idea. 



6 THE BINARY STARS 

Herschel's telescopes were more powerful than any earlier 
ones and with them he soon discovered a far larger number 
of double stars than he had anticipated. With characteristic 
thoroughness he nevertheless decided to carry out his plan of 
examining "every star in the heavens," and carefully recorded 
full details of all his observations. These included a general 
description of each pair and also estimates, or measures with 
the "revolving micrometer," or "lamp micrometer," both 
invented by himself, of the apparent distance between the two 
components and of the direction of the smaller star from the 
larger. The direction, or position angle, of the smaller star, 
by his definition, was the angle at the larger star between the 
line joining the two stars and a line parallel to the celestial 
equator. The angle was always made less than 90°, the letters, 
nf, sf, sp, and np being added to designate the quadrant. His 
first catalogue, presented to the Royal Society on Jan. 10, 
1782, contains 269 double stars, "227 of which, to my present 
knowledge, have not been noticed by any person." A second 
catalogue, containing 434 additional objects, was presented to 
the same society in 1784. The stars in these catalogues were 
divided into six classes according to angular separation. 

In the first [he writes] I have placed all those which require indeed a 
very superior telescope, the utmost clearness of air, and every other 
favorable circumstance to be seen at all, or well enough to judge of 
them. ... In the second class I have put all those that are proper for 
estimations by the eye or very delicate measures of the micrometer. . . . 
In the third class I have placed all those . . . that are more than five 
but less than 15" asunder; . . . The fourth, fifth, and sixth classes 
contain double stars that are from 15" to 30", from 30" to 1' and from 
1' to 2' or more asunder. 

Class I, in the two catalogues, includes 97 pairs, and contains 
such systems as r Ophiuchi, 5 Herculis, e Bootis, £ Ursae Majoris, 
4 Aquarii, and f Caned. In general, Herschel did not attempt 
micrometer measures of the distances of these pairs because 
the finest threads available for use in his micrometers sub- 
tended an angle of more than 1". The following extracts 
will show this method of estimating the distance in such cases 
and of recording the position angle, and also the care with which 
he described the appearance of each object. The dates of 
discovery, or of the first observation, here printed above the 
descriptions, are set in the margin at the left in the original. 



HISTORICAL SKETCH: THE EARLY PERIOD 7 

H. 1. September 9, 1779 

€ Bootis, Flamst. 36. Ad dextrum femur in perizomate. Double. 
Very unequal. L. reddish ; S. blue, or rather a faint lilac. A very beauti- 
ful object. The vacancy or black division between them, with 227 is 
% diameter of S.; with 460, 1% diameter of L.; with 932, near 2 diame- 
ters of L.; with 1,159, still farther; with 2,010 (extremely distinct), 
% diameters of L. These quantities are a mean of two years' observa- 
tion. Position 31° 34' n preceding. 

H. 2. May 2, 1780 

£ Ursae Majoris. Fl. 53. In dextro posteriore pede. Double. A 
little unequal. Both w [white] and very bright. The interval with 222 
is % diameter of L.; with 227, 1 diameter of L; with 278, near 1)4 
diameter of L. Position 53° 47' s following. 

Careful examination of the later history of the stars of Her- 
schel's Class I shows that the majority had at discovery an 
angular separation of from 2" to 3"5; a half dozen pairs as 
wide as 5" are included (one with the manuscript remark, "Too 
far asunder for one of the first class"); and a number as close 
as or closer than 1". Seven of these stars do not appear in the 
great catalogue of Struve, but five of these have been recovered by 
later observers, leaving only two that cannot be identified. 

In passing judgment upon the accuracy, or the lack of it, in 
Herschel's measures of double stars, it is necessary to hold in 
mind the conditions under which he had to work. His reflec- 
tors (all of his own construction) were indeed far more powerful 
telescopes than any earlier ones, especially the "twenty-feet 
reflector," with mirror of 18%-in. aperture, and the great 
"forty-feet telescope," with its 4-ft. mirror. But these telescopes 
were unprovided with clockwork; in fact, their mountings were 
of the alt-azimuth type. It was therefore necessary to move the 
telescope continuously in both coordinates to keep a star in the 
field of view and the correcting motions had to be particularly 
delicate when high-power eyepieces, such as are necessary in the 
observation of close double stars, were employed. Add the 
crude forms of micrometers at his disposal, and it will appear 
that only an observer of extraordinary skill would be able to 
make measures of any value whatever. 

No further catalogues of double stars were published by 
Herschel until June 8, 1821, about a year before his death, 



8 THE BINARY STARS 

when he presented to the newly founded Royal Astronomical 
Society a final list of 145 new pairs, not arranged in classes, 
and, for the most part, without measures. 

After completing his second catalogue, in 1784, Herschel 
seems to have given relatively little attention to double stars 
until about the close of the century and, though he doubtless 
tested it fully, there is no mention of his parallax method in 
his published writings after the first paper on the subject. A 
thorough review of his double star discoveries which he insti- 
tuted about the year 1797, with careful measures, repeated in 
some cases on many nights in different years, revealed a remark- 
able change in the relative positions of the components in a 
number of double stars during the interval of nearly 20 years 
since their discovery, but this change was of such a character 
that it could not be produced by parallax. 

We have seen that, in 1782, Herschel considered the time 
not ripe for theorizing as to the possible revolution of small 
stars about larger ones. Probably no astronomer of his own 
or of any other age was endowed in a higher degree than Herschel 
with what has been termed the scientific imagination; certainly 
no one ever more boldly speculated upon the deepest problems of 
sidereal astronomy; but his speculations were the very opposite 
of guesswork; invariably they were the results of critical analyses 
of the data given by observation and were tested by further 
observations when possible. Michell, in 1783, applied his 
earlier argument from the theory of probabilities to the double 
stars in HerschePs first catalogue and concluded that practically 
all of them were physical systems; but it was not until July, 
1802, that Herschel himself gave any intimation of holding 
similar views. On that date he presented to the Royal Society 
a paper entitled Catalogue of 500 new Nebulae, nebulous Stars, 
planetary Nebulae, and Clusters of Stars; with Remarks on the 
Construction of the Heavens, in which he enumerates "the parts 
that enter into the construction of the heavens" under 12 heads, 
the second being, "II. Of Binary sidereal Systems, or double 
Stars." In this section he gives the distinction between optical 
and binary systems quoted in my Introduction and argues as to 
the possibility of systems of the latter type under the law of 
gravitation. 

On June 9, 1803, followed the great paper in which he gave 
the actual demonstration, on the basis of his measures, that 



HISTORICAL SKETCH: THE EARLY PERIOD 9 

certain double stars are true binary systems. This paper, the 
fundamental document in the history of double stars as physical 
systems, is entitled, Account of the Changes that have happened, 
during the last Twenty-five Years, in the relative Situation of 
Double-stars; with an Investigation of the Cause to which they 
are owing. After pointing out that the actual existence of 
binary systems is not proved by the demonstration that such 
systems may exist, Herschel continues, "I shall therefore now 
proceed to give an account of a series of observations on double 
stars, comprehending a period of about twenty-five years which, 
if I am not mistaken, will go to prove, that many of them, are 
not merely double in appearance, but must be allowed to be real 
binary combinations of two stars, intimately held together by 
the bonds of mutual attraction." 

Taking Castor as his first example, he shows that the change 
in the position of the components is real and not due to any 
error of observation. Then, by a masterly analysis of every 
possible combination of motions of the Sun and the compo- 
nents in this, and in five other systems, he proves that orbital 
motion is the simplest and most probable explanation in any 
one case, and the only reasonable one when all six are considered. 
His argument is convincing, his conclusion incontrovertible, 
and his paper, a year later, containing a list of 50 additional 
double stars, many of which had shown motion of a similar 
character, simply emphasizes it. 

This practically concluded Sir William Herschel's contribu- 
tions to double star astronomy, for his list of 145 new pairs, 
published in 1821, was based almost entirely upon observations 
made before 1802. In fact, little was done in this field by any 
one from 1804 until about 1816. Sir John Herschel, in that 
year, decided to review and extend his father's work and had 
made some progress when Sir James South, who had inde- 
pendently formed similar plans, suggested that they cooperate. 
The suggestion was adopted and the result was a catalogue of 
380 stars, based upon observations made in the years 1821 to 
1823 with South's 5-ft. and 7-ft. refractors, of 3^-in. and 5-in. 
aperture, respectively. These telescopes were mounted equa- 
torially but were not provided with driving clocks. They 
were, however, equipped with micrometers in which the par- 
allel threads were fine spider lines. The value of the catalogue 
was greatly increased by the inclusion of all of Sir William 



10 THE BINARY STARS 

Herschel's measures, many of which had not before been 
published. 

Both of these astronomers devoted much attention to double 
stars in following years, working separately, however, South 
with his refractors, Herschel with a 20-ft. reflector (18-in. mirror) 
and later with the 5-in. refractor which he had purchased from 
South. They not only remeasured practically all of Sir William 
Herschel's double stars, some of them on many nights in different 
years, but they, and in particular Sir John Herschel, added a 
large number of new pairs. Indeed, so numerous were J. Her- 
schel's discoveries and so faint were many of the stars that he 
deemed some apology necessary. He says, 

... so long as no presumption a priori can be adduced why the most 
minute star in the heavens should not give us that very information 
respecting parallax, proper motion, and an infinity of other interesting 
points, which we are in search of, and yet may never obtain from its 
brighter rivals, the minuteness of an object is no reason for neglecting 
its examination. . . . But if small double stars are to be watched, it is 
first necessary that they should become known; nor need we fear that 
the list will become overwhelming. It will be curtailed at one end, by 
the rejection of uninteresting and uninstructive objects, at least as fast 
as it is increased on the other by new candidates. 

The prediction made in the closing sentence was not imme- 
diately verified; on the contrary, as late as 1905 Burnham 
included in his General Catalogue of All Double Stars within 121° of 
the North Pole every pair published as a double star, even those 
which had been rejected by their discoverers when they revised 
their lists. 

The long series of measures and of discoveries of double stars 
by Herschel and South were of great value in themselves and 
perhaps of even greater value in the stimulus they gave to the 
observation of these objects by astronomers generally, and well 
merited the gold medals awarded to their authors by the Royal 
Astronomical Society. The measures, however, are now assigned 
small weight on account of the relatively large errors of observa- 
tion due to the conditions under which they were of necessity 
made; and of the thousands of new pairs very few indeed have 
as yet proved of interest as binaries. The great majority are 
too wide to give the slightest evidence of orbital motion in the 
course of a century. 



HISTORICAL SKETCH: THE EARLY PERIOD 11 

The true successor to Sir William Herschel, the man who 
made the next real advance in double star astronomy, an advance 
so great that it may indeed be said to introduce a new period 
in its history, was F. G. W. Struve. Wilhelm Struve became 
the director of the observatory at Dorpat, Russia, in 1813, 
and soon afterward began measuring the differences in right 
ascension and in declination between the components of double 




1<\ G. W. Struve. 

stars with his transit instrument, the only instrument available. 
A little later he acquired a small equatorial, inferior to South's, 
with which he continued his work, and, in 1822, he published 
his Catalogus 795 stellarum duplicium. This volume is interesting 
but calls for no special comment because Struve's great work 
did not really begin until two years later, in November, 1824, 
when he received the celebrated Fraunhofer refractor. 

That telescope as an instrument for precise measurements 
was far superior to any previously constructed. The tube was 
13 ft. long, the objective had an aperture of nine Paris inches,* 
the mounting was equatorial and of very convenient form, 
and, best of all, was equipped with an excellent driving clock. 

♦This is Struve's own statement. Values ranging from 9M to 9.9 in. 
(probably English inches) are given by different authorities. 



12 THE BINARY STARS 

So far as I am aware, this was the first telescope employed 
in actual research to be provided with clockwork though Passe- 
ment, in 1757, had "presented a telescope to the King [of France], 
so accurately driven by clockwork that it would follow a star 
all night long." A finder of 23^-in. aperture and 30-in. focus, a 
full battery of eyepieces, and accurate and convenient microm- 
eters completed the equipment, over which Struve was pardon- 
ably enthusiastic. After careful tests he concluded that "we 
may perhaps rank this enormous instrument with the most 
celebrated of all reflectors, viz., Herschel's." 

Within four days after its arrival Struve had succeeded in 
erecting it in a temporary shelter and at once began the first 
part of his well planned program of work. His object was 
the study of double stars as physical systems and so carefully 
had he considered all the requirements for such an investigation 
and so thorough, systematic, and skillful was the execution of 
his plans that his work has served as a model to all of his suc- 
cessors. His program had three divisions : the search for double 
stars; the accurate determination of their positions in the sky 
with the meridian circle as a basis for future investigations 
of their proper motions; and the measurement with the microm- 
eter attached to the great telescope of the relative positions 
of the components of each pair to provide the basis for the 
study of motions within the system. 

The results are embodied in three great volumes, familiarly 
known to astronomers as the Catalogus Novus, the Positiones 
Mediae, and the Mensurae Micrometricae. The first contains 
the list of the double stars found in Struve's survey of the sky 
from the North Pole to —15° declination. For the purposes 
of this survey he divided the sky into zones from 73^° to 10° wide 
in declination and swept across each zone from north to south, 
examining with the main telescope all stars that were bright 
enough, in his estimation, to be visible in the finder at a distance 
of 20° from the full Moon. He considered that these would 
include all stars of the eighth magnitude and the brighter ones 
of those between magnitudes 8 and 9. Struve states that the 
telescope was so easy to manipulate and so excellent in its optical 
properties that he was able to examine 400 stars an hour; and he 
did, in fact, complete his survey, estimated to embrace the 
examination of 120,000 stars, in 129 nights of actual work in 
the period from November, 1824, to February, 1827. 



HISTORICAL SKETCH: THE EARLY PERIOD 13 

Since each star had to be chosen in the finder, then brought 
into the field of view of the large telescope, examined, and, if 
double, entered in the observing record, with a general descrip- 
tion, and an approximate position determined by circle read- 
ings, it is obvious that at the rate of 400 stars an hour, only 
a very few seconds could be devoted to the actual examination 
of each star. If not seen double, or suspiciously elongated at 
the first glance, it must, as a rule, have been passed over. Struve 
indeed definitely states that at the first instant of observation 
it was generally possible to decide whether a star was single 
or double. This is in harmony with my own experience in 
similar work, but I have never been content to turn away 
from a star apparently single until satisfied that further exami- 
nation on that occasion was useless. As a matter of fact, later 
researches have shown that Struve overlooked many pairs 
within his limits of magnitude and angular separation, and 
hence easily within the power of his telescope; but even so 
the Catalogus Novus, with its short supplement, contains 3,112 
entries. In two instances a star is accidentally repeated with a 
different number so that 3,110 separate systems are actually 
listed. Many of these had been seen by earlier observers and a 
few that had entirely escaped Struve's own search were included 
on the authority of Bessel or some other observer. 

Struve did not stop to make micrometer measures of his 
discoveries while engaged in his survey, and the Catalogus 
Novus therefore gives simply a rough classification of the pairs 
according to their estimated angular separation, with estimates 
of magnitude and approximate positions in the sky based on 
the equatorial circle readings. He rejected Herschel's classes 
V and VI, taking 32" as his superior limit of distance and divid- 
ing the stars within this limit into four classes: (1) those under 
4"; (2) those between 4" and 8"; (3) those between 8" and 16"; 
and (4) those between 16" and 32". Stars in the first class 
were further distinguished as of three grades by the use of the 
adjectives vicinae, pervicinae, and vidnissimae. The following 
lines (page 14) will illustrate the form of the catalogue, the num- 
bers in the last column indicating the stars that had been 
published in his prior catalogue of 795 pairs. 

The Catalogus Novus, published in 1827, furnished the work- 
ing program on which Struve's two other great volumes were 
based, though the Positiones Mediae includes meridian circle 



14 



THE BINARY STARS 



Nume- 
rus 


Nomen 
Stellae 


A. R. 


Decl. 


Descriptio 


Num. 
C. P. 


1 




OhO'O 


+36° 15' 


II (8.9) (9) 




2 


Cephei 316 


-0.0 


+78 45 


I (6.7) (6.7), vicinae 




3 


Andromedae 31 


-0.4 


+45 25 


II (7.8) (10) = H.II 83 


1 


4 




-0.9 


+ 7 29 


II (9), Besseli mihi non 
inventa 




5 


34 Piscium 


-1.1 


+ 10 10 


III (6) (10), etiam 
Besseli 





measures made as early as 1822, and the Mensurae Micromet- 
ricae some micrometer measures made in the years 1824 to 
1828. Micrometer work was not actively pushed until 1828 
and four-fifths of the 10,448 measures in the Mensurae were 
made in the six years 1828 to 1833. The final measures for the 
volume were secured in 1835 and it was published in 1837. 
The meridian observations were not completed until 1843, and 
the Positiones Mediae appeared nine years later, in 1852. 

The latter volume does not specially concern us here, for it 
is essentially a star catalogue, giving the accurate positions of 
the 2 (the symbol always used to designate Struve's double 
stars) stars for the epoch 1830.0. The Mensurae Micromet- 
ricae, on the other hand, merits a more detailed description, 
for the measures within it hold in double star astronomy a 
position comparable to that of Bradley's meridian measures in 
our studies of stellar proper motions. They are fundamental. 
The book is monumental in form as well as in contents, its 
pages measuring 173^ by 11 in. It is, as Lewis remarks, not to 
be taken lightly, and its gravity is not lessened by the fact that 
the notes and the Introduction of 180 pages are written in 
Latin. Every serious student of double stars, however, should 
read this Introduction carefully. 

Looking first at the actual measures, we find the stars arranged 
in eight classes, Class I of the Catalogus Novus being divided 
into three, to correspond to the grades previously defined by 
adjectives, and classes III and IV, into two each. The upper 
limits of the eight classes, accordingly, are 1, 2, 4, 8, 12, 16, 
24, and 32 seconds of arc, respectively. The stars in each class 
are further distinguished according to magnitude, being graded as 
lucidae if both components of the pair are brighter than 8.5 mag- 
nitude, and reliquae if either component is fainter than this. 



HISTORICAL SKETCH: THE EARLY PERIOD 



15 



Sir John Herschel had early proposed that the actual date 
of every double star measure be published and that it be given 
in years and the decimal of a year. About the year 1828 he 
further suggested that position angles be referred to the north 
pole instead of to the equator as origin and be counted through 
360°. This avoids the liability to mistakes inherent in Sir 
William HerschePs method. Both suggestions were adopted 
by Struve and have been followed by all later observers. Gen- 
erally the date is recorded to three decimals, thus defining the 
day, but Struve gives only two. The position angle increases 
from North (0°) through East, or following (90°), South (180°), 
and West or preceding (270°). 

The heading of the first section, and the first entry under it 
will illustrate the arrangement of the measures in the Men- 
surae Micrometricae: 

Duplices Lucidae Ordinis Primi 
Quarum distantiae inter O'-'OO et l'-'OO 



Epocha 


Amplif. 


Distant. 


Angulus 


Magnitudines 


2 


Cephei 316. 


a. = O^'O. 


5 = 78° 45' 




Major— 6.3 j 


} .ava; 


mirii 


or = 6.6 eerie 


flavior 


1828.22 


600 


0''72 


342 ?5 


6.5,7 


1828.27 


600 


0.84 


343.4 


6.5, 7 


1832.20 


600 


0.94 


339.3 


6, 6 


1832.24 


480 


0.70 


337.5 


6, 6.5 


1833.34 


800 


0.85 


344.8 


6.5, 6.5m 


Medium 1830.85 




0.810 


341.50 





The Introduction contains descriptions of the plan of work, 
the instrument, and the methods of observing, and thorough 
discussions of the observations. The systems of magnitudes 
and of color notation, the division of the stars into classes by 
distance and magnitude, the proper and orbital motions detected, 
are among the topics treated. One who does not care to read 
the Latin original will find an excellent short summary in English 
in Lewis's volume on the Struve Double Stars published in 
1906 as Vol. LVI of the Memoirs of the Royal Astronomical 
Society of London. Three or four of Struve's general conclusions 
are still of interest and importance. He concludes, for example, 



16 THE BINARY STARS 

that the probable errors of his measures of distance are some- 
what greater than those of his measures of position angle and 
that both increase with the angular separation of the components, 
with their faintness, and with the difference in their magnitudes. 
Modern observers note the same facts in the probable errors of 
their measures. In their precision, moreover, and in freedom 
from systematic errors, Struve's measures compare very favorably 
with the best modern ones. 

His observations of star colors show that when the two com- 
ponents of a pair are of about the same magnitude they are 
generally of the same color, and that the probability of color 
contrast increases with increasing difference in the brightness 
of the components, the fainter star being the bluer. Very few 
exceptions to these results have been noted by later observers. 

Finally, in connection -with his discussion of the division of 
double stars into classes by distance, Struve argues, on the 
theory of probabilities, that practically all the pairs in his first 
three classes (distance under 4" 00) and the great majority in 
his first five classes (distance less than 12") are true binary 
systems. With increasing angular separation he finds that the 
probability that optical systems will be included increases, 
especially among the pairs in which both components are as 
faint as, or fainter than, 8.5 magnitude. This again is in har- 
mony with the results from more recent investigations. 

The Russian government, in 1839, called upon Struve to build 
and direct the new Imperial Observatory at Pulkowa. Here 
the principal instruments were an excellent Repsold meridian 
circle and an equatorial telescope with an object glass of 15-in. 
aperture. This was then the largest refractor in the world, 
as the 9-in. Dorpat telescope had been in 1824. 

One of the first pieces of work undertaken with it was a 
resurvey of the northern half of the sky to include all stars as 
bright as the seventh magnitude. In all, about 17,000 stars 
were examined, and the work was completed in 109 nights of 
actual observing between the dates Aug. 26, 1841, and Dec. 7, 
1842. The immediate object was the formation of a list of 
all the brighter stars, with approximate positions, to serve 
as a working program for precise observations with the meridian 
circle. It was thought, however, that the more powerful 
telescope might reveal double stars which had escaped detection 
with the 9-in. either because of their small angular separation 



HISTORICAL SKETCH: THE EARLY PERIOD 17 

or because of the faintness of one component. This expecta- 
tion was fully realized. The survey, which after the first 
month was conducted by Wilhelm Struve's son, Otto, resulted 
in the discovery of 514 new pairs, a large percentage of which 
were close pairs. These, with Otto Struve's later discoveries 
which raised the total to 547, are known as the OS or Pulkowa 
double stars. The list of 514 was published in 1843 without 
measures, and when, in 1850, a corrected catalogue, with meas- 
ures, was issued, 106 of the original 514 were omitted because 
not really double, or wider than the adopted distance limits, 
or for other reasons. But, as Hussey says, "it is difficult effec- 
tively to remove a star which has once appeared in the lists." 
Nearly all of the 02 stars rejected because of wide separation 
have been measured by later observers and are retained in 
Hussey's Catalogue of the OS Stars and in Burnham's General 
Catalogue. 

The early period of double star discovery ended with the 
appearance of the Pulkowa Catalogue. New double stars were 
indeed found by various observers as incidents in their regular 
observing which was mainly devoted to the double stars in the 
great catalogues which have been described and especially 
to those in the S and the OS lists. The general feeling, how- 
ever, was that the Herschels and the Struves had practically 
completed the work of discovery. 

Many astronomers, in the half century from 1820 to 1870, 
devoted great energy to the accurate measurement of double 
stars; and the problem of deriving the elements of the orbit 
of a system from the data of observation also received much 
attention. This problem was solved as early as 1827, and new 
methods of solution have been proposed at intervals from that 
date to the present time. Some of these will be considered in 
Chap. IV. 

One of the most notable of the earlier of these observers was 
the Rev. W. R. Dawes, who took up this work as early as 1830, 
using a 3.8-in. refractor. Later, from 1839 to 1844, he had 
the use of a 7-in. refractor at Mr. Bishop's observatory, and 
still later, at his own observatory, he installed first a 6-in. Merz, 
than a 73^-in. Alvan Clark, and finally an 8>^-in. Clark refractor. 
Mr. Dawes possessed remarkable keenness of vision, a quality 
which earned for him the sobriquet, the eagle-eyed, and, as 
Sir George Airy says, was also "distinguished . . . by a habitual, 



18 THE BINARY STARS 

and (I may say) contemplative precision in the use of his instru- 
ments." His observations, which are to be found in the volumes 
of the Monthly Notices and the Memoirs of the Royal Astronomical 
Society, "have commanded a degree of respect which has not 
often been obtained by the productions of larger instruments." 

Another English observer whose work had great influence 
upon the progress of double star astronomy was Admiral W. H. 
Smythe, who also began his observing in 1830. His observa- 
tions were not in the same class with those of Dawes, but his 
Bedford Catalogue and his Cycle of Celestial Objects became 
justly popular for their descriptions of the double and multiple 
stars, nebulae, and clusters of which they treat, and are still 
"anything but dull reading." 

Far more important and comprehensive than that of any 
other astronomer of the earlier period after W. Struve was the 
double star work of Baron Ercole Dembowski who made his 
first measures at his private observatory near Naples in 1851. 
His telescope had an excellent object glass, but its aperture 
was only 5 in. and the mounting had neither driving clock nor 
position circles. His micrometer, although it could be rotated, 
was not provided with a circle from which the position angle 
could be read off. His procedure was to use two parallel fixed 
wires separated by a known distance. As the pair was brought 
to the first wire, he would set for position angle and measure 
with the micrometer thread the position of the primary star on 
that wire. Then, letting the star pass to the second wire by the 
diurnal motion, he would measure its position on that wire also. 
The difference of the two readings and the known distance 
between the two wires gave him the two sides of a right triangle 
from which the position angle could be computed. With this 
instrument Dembowski made some 2,000 sets of measures of 
high quality in the course of eight years, though how he managed 
to accomplish it is well-nigh a mystery to observers accustomed to 
the refinements of modern micrometers and telescope mountings. 

In 1859, he secured a 7-in. Merz refractor with circles, microm- 
eter, and a good driving clock, and, in 1862, he resumed his 
double star observing with fresh enthusiasm. His general 
plan was to remeasure all of the double stars in the Dorpat 
and Pulkowa catalogues, repeating the measures in successive 
years for those stars in which changes were brought to light. 
His skill and industry enabled him, by the close of the year 



HISTORICAL SKETCH-. THE EARLY PERIOD 19 

1878, to accumulate nearly 21,000 sets of measures, including 
measures of all of the 2 stars, except 64 which for one reason or 
another were too difficult for his telescope. About 3,000 of 
the measures pertain to the OS stars and about 1,700 to stars 
discovered by Burnham and other observers. Each star was 
measured on several different nights and for the more interesting 
stars long series of measures extending over 12 or 15 or even 
more years were secured. The comprehensive character of his 
program, the systematic way in which he carried it into execution, 
and the remarkable accuracy of his measures combine to make 
Dembowski's work one of the great contributions to double 
star astronomy. He died before his measures could be published 
in collected form, but they were later (1883-1884) edited and 
published by Otto Struve and Schiaparelli in two splendid 
quarto volumes which are as indispensable to the student of 
double stars as the Mensurae Micrometricae itself. 

Madler at Dorpat, Secchi at Rome, Bessel at Konigsberg, 
Knott at Cuckfield, Engelmann at Leipzig, Wilson and Gled- 
hill at Bermerside, and many other able astronomers published 
important series of double star measures in the period under 
consideration. It is impossible to name them all here. Lewis, 
in his volume on the Struve Stars, and Burnham, in his General 
Catalogue of Double Stars, give full lists of the observers, the 
latter with complete references to the published measures. 



CHAPTER II 
HISTORICAL SKETCH : THE MODERN PERIOD 

The feeling that the Herschels, South, and the Struves had 
practically exhausted the field of double star discovery, at 
least for astronomers in the northern hemisphere, continued 
for thirty years after the appearance of the Pulkowa Cata- 
logue in 1843. Nor were any new lines of investigation in 
double star astronomy developed during this period. Then, 
in 1873, a modest paper appeared in the Monthly Notices of 
the Royal Astronomical Society, entitled "Catalogue of Eighty- 
one Double Stars, Discovered with a Six-inch Alvan Clark 
Refractor. By S. W. Burnham, Chicago, U. S. A." 

The date of the appearance of this paper may be taken as 
the beginning of the modern period of double star astronomy, 
for to Burnham belongs the great credit of being the first to 
demonstrate and utilize the full power of modern refracting 
telescopes in visual observations; and the 40 years of his active 
career as an observer cover nearly all of the modern develop- 
ments in binary star astronomy, including the discovery and 
observation of spectroscopic binaries, the demonstration that 
the eclipsing variable stars are binary systems, and the applica- 
tion of photographic methods to the measurement of visual 
double stars. 

Within a year after the appearance of his first catalogue 
Burnham had published two additional ones, raising the num- 
ber of his discoveries to 182. At that time he was not a profes- 
sional astronomer but an expert stenographer employed as 
official reporter in the United States Courts at Chicago. He 
had secured, in 1861, a 3-in. telescope with alt-azimuth mounting, 
and, some years later, a 3%-in. refractor with equatorial mount- 
ing. "This was just good enough," he tells us, "to be of some 
use, and poor enough ... to make something better more 
desirable than ever." In 1870, accordingly, he purchased the 
6-in. refractor from Alvan Clark and erected it in a small observa- 
tory at his home in Chicago. With this instrumental equipment 

20 



HISTORICAL SKETCH: THE MODERN PERIOD 



21 



and an astronomical library consisting principally of a copy of 
the first edition of Webb's Celestial Objects for Common Telescopes, 
Burnham began his career as a student of double stars. His 
first new pair (£ 40) was found on April 27, 1870. 

The 64n. telescope, which his work so soon made famous, 
was not at first provided with a micrometer and his earliest 
list of discoveries was printed without measures. Later, posi- 
tion angles were measured, but the distances continued to be 




S. W. Burnham. 



estimated. This lack of measures by him was covered to a 
considerable extent by the measures of Dembowski and Asaph 
Hall. 

Burnham's later career was unique. He held positions in 
four observatories, the Dearborn, the Washburn, the Lick, 
and the Yerkes, and discovered double stars also with the 
26-in. refractor at the United States Naval Observatory, 
the 16-in. refractor of the Warner Observatory, and the 
9.4-in. refractor at the Dartmouth College Observatory. In 
all, he discovered about 1,340 new double stars and made many 
thousands of measures which are of inestimable value because 
of their great accuracy and because of the care with which he 
prepared his observing programs. And yet, except for the two 



22 THE BINARY STARS 

short periods spent respectively at Madison and at Mount 
Hamilton, he continued his work as Clerk of the United States 
District Court of Chicago until 1910! He retired from the 
Yerkes Observatory in 1912 and died in 1921. 

Burnham's plan in searching for new double stars was very 
different from that followed by his great predecessors. He 
did not attempt a systematic survey of the sky but examined 
the stars in a more random way. In his earlier work, while 
identifying the objects described in Webb's book, he made a 
practice of examining the other stars near them. Later, when- 
ever he measured a double star, he continued this practice, 
examining in this manner probably the great majority of the 
naked eye and brighter telescopic stars visible from our latitudes. 
Many of the double stars he discovered with the 6-in. refractor 
are difficult objects to measure with an aperture of 36 inches. 
They include objects of two classes almost unrepresented in the 
earlier catalogues: pairs in which the components are separated 
by distances as small as 0"2, and pairs in which one component 
is extremely faint, and close to a bright primary. In his first 
two lists he set his limit at 10", but later generally rejected pairs 
wider than 5". The result is that the percentage of very close 
pairs, and therefore of pairs in comparatively rapid orbital motion, 
is far higher in his catalogue than in any of the earlier ones. 

Burnham's work introduced the modern era of double star 
discovery, the end of which has not yet come. No less dis- 
tinguished an authority than the late Rev. T. W. Webb, in 
congratulating Burnham upon his work in 1873, warned him 
that he could not continue it for any great length of time for 
want of material. Writing in 1900, Burnham's comment was: 
"Since that time more than one thousand new double stars 
have been added to my own catalogues, and the prospect of 
future discoveries is as promising and encouraging as when 
the first star was found with the six-inch telescope." 

Working with the 183^-in. refractor of the Dearborn Observa- 
tory, G. W. Hough discovered 648 double stars in the quarter 
century from 1881 to 1906. In 1896 and 1897, T. J. J. See, 
assisted by W. A. Cogshall and S. L. Boothroyd, examined the 
stars in the zone from —20° to —45° declination, and in half 
of the zone (from 4 h to 16 h R. A.) from -45° to -65° declination 
with the 24-in. refractor of the Lowell Observatory, and dis- 
covered 500 new double stars. See states that not less than 



HISTORICAL SKETCH: THE MODERN PERIOD 23 

10,000 stars were examined, "many of them, doubtless, on 
several occasions." This is probably an overestimate for it 
leads to a remarkably small percentage of discoveries. 

In England, in 1901, the Rev. T. E. H. Espin began pub- 
lishing lists of new double stars discovered with his 17}i-in. 
reflector. 1 The first list contained pairs casually discovered in 
the course of other work; later, Mr. Espin undertook the sys- 
tematic observation of all the stars in the Bonn Durchmusterung 
north of +30°, recording, and, as far as possible measuring, 
all pairs under 10" not already known as double. Since 1917, 
W. Milburn has been his assistant in this work, which is not 
yet completed. In 1932, Espin's published discoveries numbered 
2,444 and Milburn's, 673. 

Shorter lists of discoveries have been published by E. S. 
Holden, F. Kustner, H. A. Howe, O. Stone, Alvan and A. G. 
Clark, E. E. Barnard, and others, and many doubles were first 
noted by the various observers participating in the preparation 
of the great Astronomische Gesellschaft Catalogue. 

In France, in 1909, Robert Jonckheere began double star work 
at the Observatory of the University of Lille at Hem and in the 
course of five years discovered 1,319 new pairs. Forced by 
the war to give up his work in France, he went to the Royal 
Observatory at Greenwich, England, and for some years con- 
tinued his observations with the 28-in. refractor. The majority 
of his double stars, though close, are quite faint, a large per- 
centage of them falling outside of the 9.5 magnitude limit of 
the Bonn Durchmusterung. In 1917 he published a catalogue 
of all double stars under 5" discovered visually in the years 
1905 to 1916 in the sky area within 105° of the North Pole. 

Many pairs, generally wider than 5" and often quite faint, 
have been found in the various sections of the Astrographic 
Catalogue and listed by Scheiner, Stein, Barton, and others. 

My own work in this field of astronomy began when I came 
to the Lick Observatory in June, 1895. At first my time was 
devoted to the measurement with the 12-in. refractor of list 
of stars selected by Prof. Barnard, and the work was done under 
his direction. Later, longer lists were measured both with 
this telescope and with the 36-in. refractor; and in selecting the 
stars for measurement I had the benefit of advice — so generously 
given by him to many double star observers of my generation — 
*A 24-in. reflector was added later. Mr. Espin died on Dec. 1, 1934. 



24 THE BINARY STARS 

from Prof. Burnham, then at the Yerkes Observatory. My 
attention was early drawn to questions relating to double star 
statistics, and before long the conviction was reached that a 
prerequisite to any satisfactory statistical study of double star 
distribution was a resurvey of the sky with a large modern 
telescope that should be carried to a definite limiting magnitude. 
I decided to undertake such a survey, and, adopting the magni- 
tude 9.0 of the Bonn Durchmusterung as a limit, began the 
preparation of charts of convenient size and scale showing every 
star in the BD as bright as 9.0 magnitude, with notes to mark 
those already known to be double. The actual work of compar- 
ing these charts with the sky was begun early in April, 1899. 

Professor W. J. Hussey, who came to the Lick Observatory 
in January, 1896, also soon took up the observation of double 
stars. His first list consisted of miscellaneous stars, but, in 
1898, he began the remeasurement of all of the double stars 
discovered by Otto Struve, including the "rejected " pairs. This 
work was carried out with such energy and skill that in 1901, 
in Volume V of the Lick Observatory Publications, a catalogue 
of the OS stars was published which contained not only Hussey's 
measures of every pair but also a complete collection of all 
other published measures of these stars, with references to the 
original publications, and discussions of the motion shown by 
the various systems. In the course of this work, Hussey had 
found an occasional new double star and had decided that 
at its conclusion he would make more thorough search for new 
pairs. In July, 1899, we accordingly combined forces for 
the survey of the entire sky from the North Pole to —22° declina- 
tion on the plan which I had already begun to put into execution; 
Hussey, however, charted also the 9.1 BD stars. Each observer 
undertook to examine about half the sky area, in zones 4° wide 
in declination. When Hussey left the Lick Observatory in 
1905 to become director of the Observatory of the University 
of Michigan, I took over his zones in addition to those assigned 
to me in our division of the work and early in 1915 completed 
the entire survey to —22° declination, as originally planned, 
between 13 h and l h in right ascension, but only to — 14° declina- 
tion in the remaining 12 h . These come to the meridian in our 
winter months when conditions are rarely satisfactory for work 
at low altitudes. Subsequently, by agreement with the observers 
at the Union Observatory, South Africa, whose work will be 



HISTORICAL SKETCH: THE MODERN PERIOD 



25 



described on a later page, I extended the survey in these 12 
hours to — 18° declination. 

The survey has resulted in the discovery of more than 4,400 
new pairs, 1,329 by Hussey, the others by me, practically all 
of which fall within the distance limit of 5". Some statistical 
conclusions based upon this material will be presented in a later 
chapter. 

It may seem that undue emphasis has been placed upon the 
discovery of double stars in this historical sketch. That a par- 
ticular star is or is not double is indeed of relatively little con- 
sequence; the important thing is to secure accurate measures 
through a period of time sufficiently long to provide the data 
for a definite determination of the orbit of the system. But 
the discovery must precede the measures, as Sir John Herschel 
said long ago; moreover, such surveys as that of Struve, the 
one recently completed at the Lick Observatory and those in 
progress in the southern hemisphere, afford the only basis for 
statistical investigations relating to the number and spatial 
distribution of the double stars. Further, the comparison of 
the distance limits adopted by the successive discoverers of 
double stars and an analysis of the actual distances of the pairs 
in their catalogues afford the most convenient measure of the 
progress made in the 140 years since Herschel began his work, 
both in the power of the telescopes available and in the knowledge 
of the requirements for advance in this field of astronomy. 

The data in the first four lines of the following table are taken 
from Burnham's General Catalogue of his own discoveries, and 
in the last two lines I have added the corresponding figures for 
the Lick Observatory double star survey. 

Percentage op Close Pairs in Certain Catalogues ob Double Stars 





Class I, 


Class II, 




Per- 




number 


number 


Sum 


centage 




of stars 


of stars 




of close 
pairs 


William Herschel, catalogue of 812 stars . 


12 


24 


36 


4.5 


Wilhelm Struve, catalogue of 2,640 stars 


91 


314 


405 


15.0 


Otto Struve, catalogue of 547 stars 


154 


63 


217 


40.0 


Burnham, catalogue of 1,260 stars 


385 


305 


690 


55.0 


Hussey, catalogue of 1,327 stars 


674 


310 


984 


74.2 


Aitken, catalogue of 3,105 stars 


1,595 


710 


2,305 


74.3 



26 THE BINARY STARS 

The increasing percentage of close pairs is, of course, due in 
part to the earlier discovery of the wider pairs, but the absolute 
numbers of the closer pairs testify to the increase of telescopic 
power in the period since 1780. If Class I had been divided into 
two subclasses including pairs under 0"50 and pairs between 
0"51 and l'.'OO, respectively, the figures would have been even 
more eloquent, for 60 per cent of the Class I pairs in the 
last two catalogues enumerated have measured distances of / '50 
or less. 

While the modern period is thus characterized by the number 
of visual binaries, and, in particular, those of very small angular 
distance discovered within it, it is still more notable for the 
development of an entirely new field in binary star astronomy. 
In August, 1889, Prof. E. C. Pickering announced that certain 
lines in the objective-prism spectrograms of £ Ursae Majoris 
(Mizar) were double on some plates, single on others, the cycle 
being completed in about 104 days.* An explanation of the 
phenomenon was found in the hypothesis that the star consisted 
of two components, approximately equal in brightness, in rapid 
revolution about their center of mass. 

If the orbit plane of such a system is inclined at a considerable 
angle to the plane of projection, the velocities in the line of sight 
of the two components will vary periodically, as is evident from 
Fig. 1; and, on the Doppler-Fizeau principle, f there will be a 
slight displacement of the lines of the spectrum of each component 
from their mean positions toward the violet end when that com- 
ponent is approaching the Earth, relatively to the motion of the 
center of mass of the system, and toward the red end when it is 
receding, relatively. It is clear from the figure that when one 
component is approaching the Earth, relatively, the other will 
be receding, and that the lines of the two spectra at such times 
will be displaced in opposite directions, thus appearing double 
on the spectrograms. Twice, also, in each revolution the orbital 
motion of the two components will evidently be directly across 
the line of sight and the radial velocity of each at these times is 
the same, and is equal to that of the system as a whole. The 
lines of the two spectra, if similar, will then coincide and appear 
single on the plates. There is no question but that this explana- 

* The true period, deduced from many plates taken with slit spectro- 
graphs, is about one-fifth of this value, a little more than 20.5 days, 
t Explained in Chap. V. 



HISTORICAL SKETCH: THE MODERN PERIOD 27 

tion is the correct one, and Mizar therefore has the honor of being 
the first star discovered to be a spectroscopic binary system. 

A moment's consideration is enough to show that if one of 
the two components in such a system is relatively faint or 
"dark," only one set of spectral lines, that produced by the 
brighter star, will appear upon the plate, but that these lines will 
be shifted periodically from their mean positions just as are 
the lines in the double spectrum of Mizar. If the plane of the 
system lies so nearly in the line of sight that each star partly 
or completely eclipses the other once in every revolution, the 




To Hie earth 



B 

Fig. 1. — A, A', A", A'" = primary star at points of maximum, minimum, and 
mean radial velocity. B, B', B", B'" = position of the companion star at the 
corresponding instants. C is the center of gravity of the system. There is no 
star or other body at this point. 

presence of the darker star may be revealed by a periodic dim- 
ming of the light of the brighter one; if the orbit plane, as will 
more commonly happen, is inclined at such an angle to the line 
of sight that there is no occultation or eclipse of the stars for 
observers on the Earth, the variable radial velocity of the brighter 
star will be the sole evidence of the existence of its companion. 

Algol (j8 Persei) is a. variable star whose light remains nearly 
constant about four-fifths of the time; but once in every two 
and one-half days it rapidly loses brightness and then in a few 
hours' time returns to its normal brilliancy. As early as 1782, 
Goodericke, the discoverer of the phenomenon, advanced the 
the ry that the periodic loss of light resulted from the partial 
ecli se of the bright star by a (relatively) dark companion. In 
No ./ ber, 1889, Prof. H. C. Vogel, who had been photographing 
the SJ. 3ctn m of the star at Potsdam, announced that this theory 
was correct, for his spectrograms showed that before light mini- 



28 THE BINARY STARS 

mum the spectral lines were shifted toward the red from their 
mean position by an amount corresponding to a velocity of reces- 
sion from the Earth of about 27 miles a second. While the star 
was recovering its brightness, on the other hand, the shift of the 
lines toward the violet indicated a somewhat greater velocity of 
approach, and the period of revolution determined by means of 
the curve plotted from the observed radial velocities was identical 
with the period of light variation. Algol thus became the second 
known spectroscopic binary star and the first of the special class 
later called eclipsing binaries. 

Within a few months two other spectroscopic binary stars 
were discovered; j8 Aurigae by Miss Maury at the Harvard 
College Observatory from the doubling of the lines in its spec- 
trum at intervals of slightly less than two days (the complete 
revolution period is 3.96 days), and a Virginis, by Vogel. The 
latter star was not variable in its light, like Algol, nor did its 
spectrum show a periodic doubling of the lines,* like Mizar and 
|8 Aurigae, but the lines of the single spectrum were displaced 
periodically, proving that the star's radial velocity varied, and 
the cycle of variation was repeated every four days, a Virginis 
is thus the first representative of that class of spectroscopic 
binary systems in which one component is relatively dark, as 
in the case of Algol, but in which the orbit plane does not 
coincide with the line of sight. It is to this class that the great 
majority of spectroscopic binary stars now known belong. 
The reader must not infer that the companion stars in systems 
of this class emit no light; the expression relatively dark may 
simply mean that the companion is two or three magnitudes 
fainter than its primary. If the latter were not present, the 
companion in many systems would be recognized as a bright 
star; even the companion of Algol radiates enough light to permit 
the secondary eclipse, when the primary star is the occulting 
body, to be detected by our delicate modern photometers. 

The story of the modern spectrograph and its revelations of 
the chemical composition of the stars and nebulae and of the 
physical conditions that prevail in them is a marvelous one, 
but this is not the place to tell it. We must limit ourselves to 
the simple statement that in the years since 1889 the spectro- 
graph has also given us a vast amount of information with 

* The secondary spectrum of a Virginia has been photographed in more 
recent years. 



HISTORICAL SKETCH: THE MODERN PERIOD 29 

regard to the radial velocities of the stars and, as a byproduct, 
with regard to spectroscopic binary systems. In this develop- 
ment the Lick Observatory has taken a leading part, for by the 
application of sound engineering principles in the design of the 
Mills spectrograph, and by patient and skilful experimental 
work extended over several years, Dr. Campbell was enabled, 
in the late 1890's, to secure an accuracy of measurement of 
radial velocity far surpassing any previously attained. The 
New Mills spectrograph, mounted in 1903, led to even better 
results, and it is now possible, in the more favorable cases, to 
detect a variation in the radial velocity even if the range is only 
1.5 km/sec* Other observers and institutions have also been 
most active and successful, and the number of known spec- 
troscopic binaries has increased with great rapidity. The First 
Catalogue of Spectroscopic Binaries, compiled by Campbell and 
Curtis to include the systems observed to Jan. 1, 1905, had 
140 entries; by Jan. 1, 1910, when Campbell prepared his Second 
Catalogue of Spectroscopic Binary Stars, the number had grown 
to 306; the Third Catalogue, compiled by Dr. J. H. Moore in 
1924 had 1,054 entries, and in December, 1931, the card catalogue 
which is kept up to date at the Lick Observatory listed 1,340 stars 
with known variable radial velocity and 122 more in which 
variation was indicated or suspected. In the vast majority of 
cases the variation in radial velocity was detected at obser- 
vatories in the United States and Canada or at the Chile Station 
of the Lick Observatory. 

The institutions that have engaged most actively in the meas- 
urement of stellar radial velocities and the consequent discovery 
of spectroscopic binary stars, are the Lick (with its branch station 
at Santiago, Chile, until 1929), the Yerkes, the Mount Wilson, 
the Dominion Astrophysical (Victoria), and the Dominion 
(Ottawa) Observatories. Other observatories in the United 
States, those at Pulkowa, Potsdam, and Bonn, in Europe, and 
the Cape Observatory, in South Africa, have also made notable 
contributions in this field. 

The discoveries of the spectroscopic binary stars are ordinarily 
credited to observatories rather than individuals because it is 
often a matter for fine discrimination to decide with whom the 

* This is six times the probable error of measurement of the best plates. 
No one, however, would announce a radial velocity of so small a range on 
the basis of only two or three plates. 



30 THE BINARY STARS 

credit for a particular discovery should rest. In general, at 
least three spectrograms are required to prove that a star is a 
spectroscopic binary star. These may all be taken and meas- 
ured by a single observer, or the three plates may be exposed 
by as many different observers in the course of carrying out 
a program of work planned by a fourth; the plates may be 
measured by one or more of the four or by others; variation in 
the radial velocity may be suspected from the second plate and 
confirmed by the third or only by a fourth or still later plate. 

Not all stars showing variable radial velocity are spectroscopic 
binary stars. When, as in the case of Mizar, or of j8 Aurigae, 
two sets of lines appear upon the spectrogram, it is certain that 
we have to do with a double star system. When only one set 
of lines appears, but this set exhibits a periodic variation, as in 
the case of a Virginis, it is almost certain that the light producing 
the spectrum comes from the brighter component of such a 
system. But in some cases — the Cepheid variable stars, for 
example — the apparent variation in radial velocity may be the 
consequence of rhythmic or pulsating motion in the atmosphere 
of a single star, rather than of the orbital motion of a component 
in a binary system. Such stars, and also those in which the 
observed variation in radial velocity is quite irregular, will be 
considered later. 

The problem of finding the elements of the orbit of a spec- 
troscopic binary from the data given by the measures of radial 
velocity was solved as early as 1891 by Rambaut, and in 1894, 
Lehmann-Filh6s published the method which has been the 
chief one used ever since. A number of other methods have 
been proposed in more recent years, some analytical, others 
graphical, and doubtless others still will be developed. This 
phase of the subject is treated in Chap. VI. 

At the present time orbits for more than 320 systems have been 
computed, a number greatly exceeding that of the visual binary 
systems with known orbits. The reason is not far to seek. 
The visual binaries are systems of vast dimensions and their 
revolution periods range from a minimum (so far as known at 
present) of 4.6 years to a maximum that is certainly greater 
than 1,000 years. Castor, for example, was one of the first double 
stars to be observed, and it was the one in which the fact of 
orbital motion was first demonstrated; but although the observa- 
tions extend from the year 1719 to date, the length of the revolu- 



HISTORICAL SKETCH: THE MODERN PERIOD 31 

tion period is still quite uncertain. The spectroscopic binary 
stars, on the other hand, are, in general, systems of relatively 
small dimensions, the revolution periods ranging from five or 
six hours, as a minimum, to a few years. The masses of the 
systems being assumed to be of the same order, the smaller 
the dimensions, the greater the orbital velocity, and the greater 
the probability of the detection of the system by means of the 
spectrograph, for the amount of the displacement of the lines in 
the spectrum is a function of the radial velocity of the star. 

Now, if the revolution of a system is accomplished in, say, 
two or three days, or even in two or three weeks, or months, 
it is possible for an observer to secure ample data for the computa- 
tion of its definite orbit in a single season. Indeed, if the spectro- 
graph is devoted to this purpose exclusively and the percentage 
of clear nights is large, a single telescope may in one season secure 
the data for the orbits of twenty or more systems. 

As in the work of their discovery, so in the computation 
of the orbits of the spectroscopic binary stars the American 
observatories are taking the lead. Inspection of the table 
of orbits given in the appendix to this volume will show that 
more than 90 per cent of the orbits have been computed' by 
astronomers at the Dominion Astrophysical Observatory, at 
Victoria, and the Dominion Observatory at Ottawa, in Canada, 
and by those at the Lick, Allegheny, Yerkes, Mount Wilson, 
and Detroit Observatories in the United States. 

While the spectroscopic binary stars have been receiving ever 
increasing attention in recent years, the visual binary stars are 
by no means being neglected. The work of measuring 1 and 
remeasuring the double stars discovered has been carried on 
enthusiastically by scores of able observers with small tele- 
scopes and with large ones. It is impossible to comment upon 
all of these or to give details of the hundreds of series of meas- 
ures they have published but I cannot refrain from referring 
here to two of the most prominent observers of the generation 
that has just passed away — G. V. Schiaparelli and Asaph Hall. 
Schiaparelli's measures are published in two quarto volumes, 
the first containing the measures made at Milan with the 8-in. 
refractor, in the years 1875 to 1885; the second, the series made 
with the 18-in. refractor at the same observatory in the interval 
from 1886 to 1900. Hall's work, carried out with the 26-in. 
refractor of the United States Naval Observatory at Washington, 



32 THE BINARY STARS 

is also printed in two quarto volumes, the first containing the 
measures made in the years 1875 to 1880; the second, those made 
from 1880 to 1891. The working lists of both observers were 
drawn principally from the Dorpat and Pulkowa catalogues, 
but include many of Burnham's discoveries and some made by 
Hough and by others. The high accuracy of their measures 
and the fact that they — and Schiaparelli in particular — repeated 
the measures of the more interesting stars year after year makes 
the work of these observers of the greatest importance. 

At present, double stars are regularly measured at two or 
three of the larger observatories of this country, at several 
important observatories in England and on the continent of 
Europe,* and by many enthusiastic amateurs in this country 
and abroad. So voluminous is the literature of the subject 
that one who wishes to trace the full record of one of the dou- 
ble stars discovered by Herschel or by Struve in the original 
sources must have access to a large astronomical library. This 
condition was recognized many years ago, and as early as 
1874 Sir John Herschel's A Catalogue of 10,300 Multiple and 
Double Stars, Arranged in Order of R. A. was published as a 
Memoir of the Royal Astronomical Society. This catalogue 
attempted merely to give a consecutive list of the known double 
stars, without measures and did not go far toward meeting 
the needs of observers or computers. The first really service- 
able compendium was that published by Flammarion in 1878, 
entitled Catalogue des fitoiles Doubles et Multiples en Mouve- 
ment relatif certain. The volume aimed to include all pairs 
known from the actual measures to be in motion; 819 systems 
are listed, each with a fairly complete collection of the published 
measures, about 14,000 in all, and notes on the nature of the 
motion. For thirty years this book formed a most excellent 
guide to observers. 

The following year, 1879, A Handbook of Double Stars, pre- 
pared by Crossley, Gledhill, and Wilson, was published in 
London — a work that had a wide circulation and that proved 
of the greatest service to students of double-star astronomy. 
It is divided into three parts, the first two giving a general 
account of double-star discoveries and methods of observing and 
of orbit computation. The third section contains a "Catalogue 

* This statement refers to the northern hemisphere; an account of the 
work in the southern hemisphere is given on later pages. 



HISTORICAL SKETCH: THE MODERN PERIOD 33 

of 1,200 double stars and extensive lists of measures." An 
appendix gives a list of the principal papers on double stars. 

In 1900, Burnham published a General Catalogue of his 
own discoveries containing a complete collection of all known 
measures of these stars with notes discussing the motion when 
such was apparent, and references to the original sources from 
which the measures were taken. This proved to be the first 
of a series of such volumes. Hussey's catalogue of the Otto 
Struve stars, to which reference has already been made, was 
published in 1901, and five years later, in 1906, Lewis's great 
volume on the Struve stars appeared. This is, in effect, a 
revision of the Mensurae Micrometricae and gives all of the 
2 stars in the order of their original numbers, disregarding the 
inconvenient division into classes. Such of the S "rejected" 
stars as have been measured by later observers are also included, 
and all or nearly all of the published measures of each pair. 
The notes give an analysis and discussion of the motions that 
have been observed, and form one of the most valuable features 
of the work, for the author had devoted many years to a com- 
prehensive study of double star astronomy in all its phases. 
In 1907, Eric Doolittle published a catalogue of the Hough 
stars, all of which he had himself reobserved, and in 1915, 
Fox included in Vol. I of The Annals of the Dearborn Observatory 
catalogues of the discoveries of Holden and of Kustner with 
a new series of measures of these stars. More recently Van 
Biesbroeck (Publications of the Yerkes Observatory, Vol. V, 
Part I, 1927) has published measures of all of Hussey's pairs, 
1,298 in all, that were within reach of observation from the 
Yerkes Observatory. Thus all of the longer catalogues of new 
double stars discovered at observatories in the northern hemi- 
sphere, except my own and some other of the very recent ones 
and those of Sir John Herschel, have now been revised and 
brought up to date, for Sir William Herschel's discoveries, 
except the very wide pairs, are practically all included in the 
Mensurae Micrometricae. 

Every one of the volumes named is most convenient for 
reference and contains information not easily to be found else- 
where; but they were all surpassed by Burnham's comprehensive 
and indispensable work, A General Catalogue of Double Stars 
vnthin 121° of the North Pole, which was published by The 
Carnegie Institution of Washington in 1906. This monu- 



34 THE BINARY STARS 

mental work consists of two parts, printed in separate quarto 
volumes. Part I contains a catalogue of 13,665 double stars, 
including essentially every pair, close or wide, within the sky 
area named, that had been listed as a double star before 1906. 
The positions, for 1880, are given, with the discovery date and 
measure or estimate. Part II contains measures, notes and 
complete references to all published papers relating to each 
pair. This catalogue proved to be a most valuable guide to 
double star observers and it was in no small measure responsible 
for the great activity in double star discovery and observation 
in the following years. 

In 1917, M. Robert Jonckheere published, in the Memoirs 
of the Royal Astronomical Society (Vol. 61) a Catalogue and 
Measures of Double Stars discovered visually from 1905 to 1916 
within 105° of the North Pole and under 5" Separation. This is, 
in effect, an extension of Burnham's General Catalogue, though 
the author excluded pairs wider than 5" instead of recording 
every pair announced by its discoverer as double and adopted 
— 15° instead of —31° -or the southern sky limit. The volume 
is particularly valuable because it gives in collected form Jonck- 
heere's own discoveries with measures at a second epoch as 
well as at the time of discovery. The other long lists in the 
volume are Espin's discoveries and those made at the Lick 
Observatory; in all, there are 3,950 entries. 

On his retirement from the Yerkes Observatory, Burnham 
turned over to Prof. Eric Doolittle, Director of the Flower 
Observatory, the material he had accumulated for a revision 
or extension of his catalogue. Doolittle set up a card catalogue 
planned to contain a complete record of measures and orbits 
published after 1906. This catalogue and a collection of books 
and pamphlets on double stars came to me after Doolittle's 
untimely death in 1920, for, in 1919, in response to his urgent 
request, I had promised to carry on the work if he did not live 
to complete it. 

The result was the publication, early in 1932, of a New General 
Catalogue of Double Stars within 120° of the North Pole.* 
Designed, in a general way, to supplement Burnham's great work, 
it gives for each pair listed in it the earliest available measure 

* New General Catalogue of Double Stars within 120° of the North Pole, 
by Robert Grant Aitken, in succession to the late Eric Doolittle, Carnegie 
Institution of Washington, Publication 417, 2 Vols., 1932. 



HISTORICAL SKETCH: THE MODERN PERIOD 35 

and all later measures, except those quoted or referred to by 
Burnham, together with appropriate notes. 

Not all pairs listed in the earlier work, however, are included 
in this new catalogue. It was thought that the wider and fainter 
pairs could to advantage be omitted, and limits were therefore 
set, based upon the apparent magnitude and the angular separa- 
tion of the components. These limits are defined by the equation 

log p = 2.8 - 0.2m, 

in which p is the angular separation and m the apparent magni- 
tude. The constant, 2.8, sets the limit at 10" for a pair whose 
apparent magnitude is 9.0. Although approximately three 
out of every ten pairs listed by Burnham are excluded by these 
limits, so numerous have been the later discoveries that the new 
catalogue has 17,181 entries as compared with Burnham's 
13,665. The catalogue includes all measures known to me that 
were made prior to 1927.0. A card catalogue of measures 
published later is kept up to date at the Lick Observatory. 

It has been convenient in this narrative to confine attention 
to this point to the double star work done at observatories in 
the northern hemisphere, for, until quite recent years it was 
there that this branch of astronomy received most attention. 
Now, however, conditions are changed and the most active and 
fruitful work in the discovery and measurement of double stars is 
that carried on at the observatories in South Africa. 

We have seen that two of the earliest double stars discov- 
ered — a Centauri and a Cruris — were stars not visible from 
European latitudes; but the first extensive list of double stars 
collected at a southern observatory was James Dunlop's cata- 
logue of 253 pairs observed at Parametta, N.S.W., in the years 
1825 to 1827 with a 9-ft. reflecting telescope. These stars, 
however, are as a rule very wide pairs and are of comparatively 
little interest. A few double stars are contained in Brisbane's 
Parametta catalogue, published in 1835, and more in the later 
meridian catalogues of the Royal Observatory at the Cape of 
Good Hope, the Argentine National Observatory at Cordoba, 
and of other southern observatories. 

The most important early paper on southern double stars 
is beyond question the chapter upon them in Sir John Herschel's 



36 THE BINARY STARS 

Results of Astronomical Observations made during the Years 
1834, 1835, 1836, 1837, 1838 at the Cape of Good Hope which 
was published in 1847. Innes says, "The sections on double 
stars in this work are to the southern heavens what Struve's 
Mensurae Micrometricae are to the northern heavens." A 
catalogue is given of the discoveries made at Feldhausen, C. G. H., 
with the 20-ft. reflector which contains the pairs h3347 to 
h 5449, together with measures of such previously known 
pairs as were encountered in the "sweeps." Many of the new 
pairs are wide and faint, resembling the h stars discovered 
at Slough, in England; but many others are comparatively 
close, many are very bright, and a number are among the finest 
double stars in the southern sky. Another division of this 
chapter gives the micrometer measures, made with the five-inch 
refractor, of many of these new pairs and of some of the known 
ones. Innes says that "the angles of the pairs are all through 
of high excellence"; but Herschel himself points out the sources 
of weakness in his measures of distances. 

Herschel's station at Feldhausen was not a permanent obser- 
vatory, and when he returned to England work there was 
discontinued; nor was double star work seriously pursued at 
any other southern station until about 40 years later. In 
1882, a list of 350 new pairs was published by H. C. Russell, 
director of the Sydney Observatory, N.S.W., the measures 
being made by Russell and by L. Hargrave. In 1884, an addi- 
tional list of 130 pairs, mostly wide, was published, and in 
the following years several lists of measures by these observers 
and their colleague, J. A. Pollock, a few of the measured pairs 
being new. In 1893, R. P. Sellors published a short list (14 pairs, 
all under 2") discovered by him at the same observatory, and 
in the following years he contributed many measures of known 
pairs and discoveries of a few additional new ones. 

A new chapter in the history of double star astronomy in the 
southern hemisphere was opened in 1896, when Dr. R. T. A. Innes 
joined the staff of the Royal Observatory at the Cape of Good 
Hope. Innes had already published as "probably new" two 
short lists of 26 and 16 stars, respectively, discovered at Sydney, 
N.S.W., in 1895 and 1896, with a 6>i-in. refractor and a small 
reflector. At the Cape Observatory, in addition to work in 
other lines, he continued his double star observing with the 
7-in., and later with the 18-in. (McClean) refractor. With these 



HISTORICAL SKETCH: THE MODERN PERIOD 37 

instruments he brought the total of his discoveries to 432 and 
made fine series of measures of these and of other southern 
double stars. In 1903 he became Government Astronomer at 
the Union Observatory, Johannesburg, South Africa. Here 
he worked with a 9-in. refractor until April, 1925, when the 
26J^-in. Grubb refractor, ordered some years before the war 
began, was finally installed. 

In August, 1925, Dr. W. H. van den Bos, who had been doing 
excellent double star work at Leiden, came to the Union Obser- 
vatory. A plan, which Innes had long cherished, for a systematic 
survey of the southern sky along the general lines of the one 
of the northern sky carried out at the Lick Observatory, was 
immediately put into execution. The major part of this survey 
has been carried out by van den Bos, though Dr. W. S. Finsen 
and other assistants in the observatory (as well as Innes himself 
until he retired in 1927) have participated. The survey is still 
in progress but is nearing completion. Including the earlier 
work at the Union Observatory by Innes and others, fully 
4,000 double stars had been discovered before the close of the 
year 1931. Innes' own discoveries total 1,613, Finsen's 300, 
while those of van den Bos exceed 2,000. These are all close 
pairs, comparable in every respect with those discovered at the 
Lick Observatory. 

In 1911, Hussey accepted the directorship of the observatory 
of the La Plata University, Argentina, in addition to his duties 
at Ann Arbor, Michigan. During his periods of residence at 
La Plata he used the 17-in. refractor in searching for and measur- 
ing southern double stars. His discoveries there brought 
his total number of new pairs up to 1,650, and measures of these 
later pairs were promptly published. 

Mr. Bernhard H. Dawson assisted Prof. Hussey at La Plata 
until 1917, when Hussey resigned the directorship, and has 
since continued to give part of his time to double star work. 
Hussey now found it possible to plan for further double star 
work in the southern hemisphere with a more powerful telescope, 
the funds for the construction of which had been provided by 
his friend and college classmate, Mr. R. P. Lamont. A 27-in. 
lens was ordered from the John A. Brashear Company, and 
Hussey personally supervised the designs for the mounting. 
Delays were encountered, however, and the war came on, with 



38 THE BINARY STARS 

the result that the telescope was not ready for use until the 
summer of 1926. 

Professor Hussey, in 1924, had personally selected a site at 
Bloemfontein, South Africa, for his southern station, and was 
on his way there to supervise the erection of the telescope and to 
carry on double star work when he died of heart disease, in Lon- 
don, on Oct. 23, 1926. The telescope, however, was erected in 
accordance with his plans, Prof. R. A. Rossiter being placed in 
charge, with Morris K. Jessup and Henry F. Donner as assistants. 

Arrangements were made with the Union Observatory for 
cooperative work in prosecuting the survey of the southern 
heavens, with gratifying results. By October, 1931, 4,712 new 
double stars were discovered and measured on one or more 
nights; 1,961 by Rossiter, 1,424 by Jessup, and 1,327 by Donner. 
These measures, for the most part, are as yet unpublished.* 

At the Union Observatory, all stars as bright as 9.0 magnitude 
in the Cape Photographic Durchmusterung are examined, as well 
as those of fainter photographic magnitude which are estimated 
to be as bright as 9.0 visual magnitude. The distance limit for 
pairs listed as double stars is set by the curve log p = 2.5 — 0.2m, 
which gives 5"0 for a pair of 9.0 magnitude. 

At the Lamont Hussey Observatory, the survey is being carried 
to all stars as bright as 9.5 in CPD, and the distance limit is 
that given by the curve log p = 2.625 — 0.2m, giving 6"75 at 
9.0 magnitude, and pairs even wider (to the limit set by 
log p = 2.8 — 0.2m) are retained. The result is that while 
3,206 pairs fall within the limits of the Union Observatory curve, 
the percentage of pairs fainter than 9.0 and comparatively wide 
is very high. 

Any statistical discussion of the number and distribution of 
double stars based upon the material that will be available after 
these southern surveys are completed, must take account of 
the systematic difference between visual and photographic 
magnitudes and of those between the various systems of visual 
magnitudes. To be significant, such a discussion must include 

* Rossiter's first list of measures of the pairs of his own discovery has since 
been published in the Memoirs of the Royal Astronomical Society, Vol. LXV, 
Part II, 1933. In his introductory note he states that his discoveries to 
date exceed 2,350 pairs. Measures only of those found prior to October, 
1932, however, a total of 2,232 pairs, are included in his paper. The list 
is notable for the large number of very close faint pairs contained in it. 



HISTORICAL SKETCH: THE MODERN PERIOD 39 

the double stars in both hemispheres and should be based upon 
accurate photometric magnitudes. 

Innes, in 1899, published a Reference Catalogue of Southern 
Double Stars, designed to include "all known double stars having 
southern declination at the equinox of 1900." The author, 
however, set limits denning the pairs of stars to be regarded 
as "double stars," the limits ranging from 1" for pairs of the 
ninth magnitude to 30" for those of the first magnitude. In 
principle, this procedure is correct, but Innes* actual narrow 
limits are open to criticism. He abandoned these limits when, 
with the assistance of Dawson and van den Bos, he compiled 
his loose-leaf catalogue that was planned to contain every 
known double star within the limits of the Cape Photographic 
Durchmusterung {i.e., south of —19° at the equinox of 1875.0) 
"that had been measured on more than one occasion as well as 
many measured only once." This loose-leaf catalogue was 
completed in 1927, but it was regarded as merely a temporary 
guide to southern observers, the declared intention being to 
issue a complete catalogue in more permanent form after the 
termination of the survey of the southern skies, initiated in 1925. 
Meanwhile, a card catalogue of all discoveries and published 
measures is kept up to date at the Union Observatory. 

Our knowledge of the spectroscopic binary stars in the far 
southern skies rests almost entirely upon the work carried on 
at the D. O. Mills Station of the Lick Observatory, established 
at Santiago, Chile, in 1903, and maintained until April, 1929, 
when it was sold to the Catholic University of Chile. The 
instrumental equipment consisted of a 37^-in. silver-on-glass 
reflector and spectrographs similar in design to those in use on 
Mount Hamilton. The working program was the measurement 
of the radial velocities of the stars and nebulae which are too 
far south to be photographed at the Lick Observatory itself. 
The discovery of binary stars was not the object in view, but a 
large percentage of the entire number of these systems known 
at the present time were found at this Station in the 26 years 
of its existence. When we add to this number the spectroscopic 
binary stars with southern declinations which have been detected 
by observers at stations in the northern hemisphere, we shall 
find that in this field there is little or no disparity between the 
two hemispheres of the sky. 



40 THE BINARY STARS 



References 

In addition to general accounts of the binary stars in standard textbooks 
on astronomy, reference may be made to two recent publications: 
Henkoteatj, F. C. : "Double and Multiple Stars," Handbuch der Astrophysik, 

Band VI, Chap. 4, pp. 299-474, Berlin, 1928. 
Baize, P.: L'Astronomie des fitoiles Doubles, Bull. Soc. Astron. de France 

44, 268, 359, 395, 505; 46, 21, Paris, 1930-1931. 

[Dr. Hamilton M. Jeffers and Mrs. Frances Greeby at Lick Observatory have 
been keeping up visual binary orbit data. Their new general catalogue is now 
ready for publication. It lists about 65,000 double stars, and indicates each 
pair for which an orbit has been calculated. North of —20°, the compilation 
has been made at Lick Observatory. South of —20°, the catalogue of Dr. 
Willem H. van den Bos is incorporated. Mrs. Greeby plans to keep up the 
catalogue of observations, and additions and changes to the general catalogue. 

Attention should also be called to the new catalogue (to be published) by 
Charles Worley of the Naval Observatory. — J.T.K.] 



CHAPTER III 

OBSERVING METHODS, VISUAL BINARY STARS 

The operation of measuring a double star is a very simple 
one. The object is to define at a given instant the position of 
one star, called the companion, with respect to the other, known 
as the primary. When the two stars are of unequal magnitude 
the brighter is chosen as the primary; when they are of equal 
brightness, it is customary to accept the discoverer's designations. 

From the first work by Sir William Herschel, the measures 
have been made in polar coordinates; and since about 1828, 
when Sir John Herschel recommended the practice, the posi- 
tion angle has been referred to the North Pole as zero point 
and has been counted through 360°. 

That is, the position angle is the angle at the primary star 
between the line drawn from it to the North Pole and one 
drawn from it to the companion, the angle increasing from 
zero when the companion is directly north, through 90° when 
it is at the east, 180° when it is south, 270° when it is west, 
up to 360° when it is once more directly north. The distance 
is the angular separation between the two stars measured at right 
angles to the line joining their centers. The two coordinates 
are usually designated by the Greek letters and p, or by the 
English letters p and s. 

THE MICROMETER 

The filar or parallel-wire micrometer is the instrument now in 
almost universal use for visual measurements of double stars.* 

* Mr. F. J. Hargreaves has recently perfected a comparison-image microm- 
eter which he finds more accurate than the filar micrometer, particularly 
in the measurement of angular distances. Two artificial star images are 
projected into the field of view side by side with the images of the double 
star to be measured. The artificial star images may be made comparable 
to those of the real stars in both color and brightness and their position 
angle and angular separation may be brought into accurate agreement 
with those of the double star. A full description is given in the Monthly 
Notices, R.A.S. (92, 72, 1931), but the instrument has not yet come into 
general use. 

41 



42 THE BINARY STARS 

A complete description of it is not necessary here; for this, the 
reader is referred to Gill's article on the Micrometer in the 
Encyclopaedia Britanniea (9th ed,), in which other forms are also 
described. Essentially it consists of a tube or adapter firmly 
fitted into the eye end of the telescope and carrying on its outer 
end a graduated circle (the position circle) reading from 0° to 
360° in a direction contrary to the figures on a clock dial. A 
circular plate fitting closely within the position circle and 
adjusted to turn freely within it carries an index, or a vernier, 




Plate II. — The micrometer for the 30-in. refractor, Lick Observatory. 

or both, to give the circle reading. In the micrometers in 
use at the Lick Observatory, this plate is rotated about the 
optical axis of the telescope by an arm carrying a pinion which 
meshes into rack teeth cut on the outer circumference of the 
position circle, A clamp is provided to hold the plate and 
circle together at any desired reading, and a tangent screw 
to give a slow motion. Upon the vernier plate an oblong box 
is mounted within which the parallel wires or threads (they 
are usually spider lines) are placed. This box is movable 
longitudinally by a well-cut, but not very fine screw. One 
thread, the fixed thread, is attached to the inner side of the 
upper plate of the box, and the other, the micrometer or mov- 
able thread, is attached to a frame or fork which slides freely 



OBSERVING METHODS, VISUAL BINARY STARS 43 

in the box longitudinally, but without any lateral play. The 
fork is moved by a very fine and accurately cut screw which 
enters the box at one end. At its outer extremity, this screw 
carries a milled head divided into 100 parts, the readings increas- 
ing as the screw draws the micrometer thread toward the head. 
Strong springs at the opposite end of the fork carrying this 
thread prevent slack or lost motion. 

The two threads, the fixed and the micrometer, must be so 
nearly in the same plane — the focal plane of the objective — 
that they can be brought into sharp focus simultaneously in an 
eyepiece of any power that may be used, but at the same time 
must pass each other freely, without the slightest interference. 
Instead of a single .fixed thread, some micrometers carry sys- 
tems of two, three, or more fixed threads, and frequently also 
one or more fixed transverse threads. Some also substitute 
two parallel threads separated a few seconds of arc for the 
single movable thread. For double star work, the simple 
micrometer with only two threads is unquestionably to be 
preferred, and even for comet, asteroid, satellite, and other 
forms of micrometric work I regard it as superior to the more 
complicated forms and less liable to lead to mistakes of record. 
Not all observers, however, will agree with me on this last point. 

The telescope is assumed to be mounted stably and to be in 
good adjustment. Assured as to these two points and as to the 
firm attachment of the micrometer to the telescope tube so that 
the zero reading of his position circle shall remain constant, 
the double star observer has still to determine the value of one 
revolution of his micrometer screw and the zero or north point 
reading of his position circle before beginning actual measure- 
ments. The reading for coincidence of the threads is elimi- 
nated by the method of double-distance measures, as will be 
shown presently, and the distances themselves are, in general, 
so small, and modern screws so accurate, that irregularities 
in the screw and corrections for temperature may be regarded 
as negligible. If desired, however, they may be determined 
in connection with measures for the revolution value. 

THE ZERO POINT 

The determination of the zero point will be considered first. 
The simplest practical method, and the one adopted by observers 
generally, is to put on the lowest power eyepiece that utilizes 



44 THE BINARY STARS 

the entire beam of light, direct the telescope upon an equatorial 
star near the meridian, stop the driving clock, and turn the 
micrometer by the box screw and the position-circle pinion 
until the star "trails" along the thread across the entire field 
of view. The star should be bright enough to be seen easily 
behind the thread, but not too bright. With the 12-in. telescope 
I find a star of the seventh or eighth magnitude most satisfactory; 
with the 36-in. telescope, one of the ninth or tenth magnitude. 
A little practice will enable the observer to determine his "par- 
allel" reading with an uncertainty not greater than one-fifth 
of one division of his circle. On the micrometer used with the 
36-in. telescope, this amounts to 0-05. Several independent 
determinations should be made. If the micrometer is not 
removed from the telescope and is set firmly to the tube, it is 
probable that the parallel reading need be checked only once or 
twice a week. When, as at the Lick Observatory, the micrometer 
is liable to be removed almost any day and is certainly removed 
several times every week, the observer very promptly forms the 
habit of determining the parallel at the beginning of his work 
every night; my own practice is to check the value at the close 
of work also. 

Ninety degrees added to the parallel gives the north point or zero 
reading. 

REVOLUTION OF THE MICROMETER SCREW 

The value of one revolution of the micrometer screw should 
be determined with the greatest care and the investigation 
should be repeated after a reasonable time interval to detect 
any wear of the screw. Two different methods of procedure 
are about equally favored by observers: the method of transits 
of circumpolar stars and the method of direct measures of the 
difference in declination of suitable pairs of stars. 

In the first method the position circle is set for the zero reading 
(i.e., 90° from the reading for parallel) and the telescope turned 
upon the star a short time before it culminates. (The driving 
clock, of course, is stopped.) Set the micrometer thread just 
in advance of the star as it enters the field of view (it is convenient 
to start with the milled head set at zero of a revolution) and note 
the time of the star's transit either on the chronograph or by 
the eye-and-ear method. Advance the thread one revolution 
or a suitable fraction of a revolution and take another transit, 



OBSERVING METHODS, VISUAL BINARY STARS 45 



and repeat this procedure until the star has crossed the entire 
field of view. A low-power eyepiece should be used and the 
series of measures so planned that they will extend over from 
forty to eighty revolutions of the screw, about half of the transits 
being taken before the star crosses the 
meridian, the other half after. Great 
care must be taken not to disturb the 
instrument during the course of the 
observations for the slightest changes 
in its position will introduce errors 
into the measures. It is well to re- 
peat the observations on a number of 
nights, setting the telescope alternately 
east and west of the pier. A sidereal 
timepiece should be used in recording 
the time of transits and if it has a large 
rate, it will be necessary to take this 
into account. 

In Fig. 2, let P be the pole, EP the observer's meridian, 
ab the diurnal path of a star, AS the position of the micrometer 
thread when at the center of the field and parallel to an hour 
circle PM, and BS' any other position of the thread. Now 
let Wo be the micrometer reading, to the hour angle, and To 
the sidereal time when the star is at S, and m, t, and T the cor- 
responding quantities when the star is at S', and let R be the 
value of one revolution of the screw. 

Through S' pass an arc of a great circle S'C perpendicular to 
AS. Then, in the triangle CS'P, right-angled at C, we have 

CS' = (m - m )R, S'P = 90° - 5, CPS' = t - to = T - T 




Fig. 2. 



and we can write 

sin [(w — m )R] = sin (T — T ) cos 8 

or, since (wi — m^)R is always small, 

cos 6 
(m - m )R = sin (T - To) -r-jry 

Sill X 



Similarly, for another observation, 

{m' - m )R = sin (T' - To) 



cos 8 
shTT 7 



(1) 



(2) 



(3) 



46 THE BINARY STARS 

Combining these to eliminate the zero point, 
(m' - m)R = sin {T - T ) ^p - sin (T - To) ^p (4) 

from which the value of R is obtained. The micrometer readings 
are supposed to increase with the time.* 

If 80 transits have been taken, it will be most convenient 
to combine the first and the forty-first, the second and the 
forty-second, and so on, and thus set up 40 equations of condi- 
tion of the form of Eq. (4). The solution of these equations by 
the method of least squares will give the most probable value 
for R. The value of R given by Eq. (4) must be corrected for 
refraction. It will suffice to use the approximate formula 

dR = -R tan 1" cot (5 - <p)r (5) 

where r is the mean refraction, 5, the declination of the star, and <p, 
the latitude of the observer. If a star is observed at lower cul- 
mination, 5 must be replaced by (180° — 8). 

If the value of R is to be detennined by direct measures of 
the difference of declination between two stars, the stars should 
satisfy the following conditions: they should lie on, or very 
nearly on, the same hour circle; their proper motions as well 
as their absolute positions at a given epoch should be accurately 
known; they should be nearly of the same magnitude and, 
if possible, of nearly the same color; the difference of declina- 
tion should amount to from fifty to one hundred revolutions 
of the micrometer screw; and, since this will ordinarily exceed 
the diameter of the field of view of the eyepiece, one or more 
intermediate stars (whose positions do not need to be so accu- 
rately known) should lie nearly on the line joining them and at 
convenient intervals to serve as steps. 

There are not many pairs of stars which answer all of the 
requirements. Probably the most available ones are to be 
found in the Pleiades and other open clusters which have been 
triangulated by heliometer observations. 

The measures should be made only on the most favorable 
nights and at times when the stars are high enough in the sky 
to make the correction for refraction small. The difference 
of declination should be measured from north star to south 
star and also in the opposite direction and the measures should 

* From Campbell's Practical Astronomy. 



OBSERVING METHODS, VISUAL BINARY STARS 47 

be repeated on several nights. If extreme accuracy is desired 
in the refraction corrections the thermometer and barometer 
should be read at the beginning and also at the end of each 
set of measures, and if the effect of temperature is to be included 
in the determination of R, measures must be made at as wide a 
range of temperature as is practicable. 

In making the reductions, the star places are first brought 
forward from the catalogue epoch to the date of the actual 
observations by correcting rigorously for precession, proper 
motion, and the reduction from mean to apparent place. The 
apparent place of each star must then be corrected for refrac- 
tion. It will generally be sufficiently accurate to use Com- 
stock's formula, in the following form : 

Refraction in 8 = — ... , . tan z cos q (6) 

4oU -+- 1 

where z is the apparent zenith distance, and q the parallactic 
angle of the star, b the barometer reading in inches and t the 
temperature of the atmosphere in degrees Fahrenheit. In 
practice I have found it more convenient to correct each star 
for refraction in the manner described than to correct the differ- 
ence in declination by the use of differential formula. 

The following pairs of stars in the Pleiades have actually 
been used by Prof. Barnard in determining the value of one 
revolution of the micrometer screw of the 40-in. telescope of 
the Yerkes Observatory: 

BD Mag. BD Mag. AS 

+ 23°537 (7.5) and + 23°542 (8.2) 696"l9 

+ 23°516(4.8) and + 23°513 (9.0) 285.94 

+ 23°557(4.0) and + 23°559 (8.4) 599.58 

+ 23°561 (7 . 5) and + 23°562 (7.8) 479 . 1 1 

+ 23°558(6.2) and + 23°562 (7.8) 401.10 

+ 23°563 (7 . 2) and + 23°569 (7 .5) 494 . 14 

+ 23°557(4.0) and +23°558(6.2) 300.25 

+ 23°507(4.7) and + 23°505(6.5) 633.40 

The differences in declination given in the final column are 
for the epoch 1903.0 and are the results of Dr. Elkin's measures 
with the Yale heliometer. 

The last pair in the list consists of the bright stars Electro, and 
Celaeno, and the table that follows gives the measures of them 
made by Barnard, in 1912, to determine the screw value for the 



48 



THE BINARY STARS 



o 
08 " 



CQ 



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bC 

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3 d 

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II 









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HQIQNQOIQ 



CO CO CO CO CO 



88 



0000000 

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m cm cm' cm e* ei cm 



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+++++++ 



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eo c5 cm ^ i-h •* 

"Si C* CM i-h H w CM 

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10 eo ■<* c« eo (N 

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OBSERVING METHODS, VISUAL BINARY STARS 49 

micrometer of the 40-in. Yerkes refractor. Step stars of magni- 
tude 11.0 and 11.5, respectively, lying nearly in the line joining 
the two bright stars were used to reduce the intervals actually 
measured. Both the tube of the 40-in. telescope and the screw 
of the micrometer are of steel and therefore mutually correct 
each other in temperature changes, at least approximately; 
but the focal length of the object glass is % in. shorter in winter 
than in summer whereas the tube shortens only 3^ in. A 
slight correction is therefore necessary if all of the measures 
are to be reduced to the focus for a common temperature. 
The column Scale reading, in the table of measures, gives the 
readings for focus on the draw tube of the telescope and the 
following column, the corrections required to reduce the meas- 
ures to the focal length corresponding to a temperature of 
60°F. The remaining columns are self-explanatory. 

If a suitable measuring engine, like those used in measuring 
positions of images on a photographic plate, is available, it may 
be used to measure the value of the revolution of the micrometer 
screw in terms of millimeters, provided the pitch of the screw 
is known accurately. Dividing the result by the focal length 
of the telescope, in millimeters, and multiplying by the value 
of the radius of a circle expressed in seconds (206,264.8) will 
reduce it to seconds of arc. Burnham's original measure of the 
screw value of the 36-in. micrometer of the Lick Observatory 
was #'907 ± 0"006. This was based on measures of the differ- 
ence in declination of two stars, seven different pairs being meas- 
ured. Wright has recently measured the screw under one of 
our measuring engines, finding 9"9045 ± 0'/0005, the value 
corresponding to a temperature of 62° F. 

MEASURING A DOUBLE STAR 

When the telescope has been directed upon the star and 
clamped, the star is brought up to the threads by means of the 
screw moving the entire micrometer box. The position angle 
is then measured, and in doing this my practice is to run the 
micrometer thread well to one side of the field of view, bring 
the double star up to the fixed thread by means of the screw 
moving the box and then rotate the micrometer by means of 
the pinion provided, keeping, meanwhile, the fixed thread 
upon the primary star, until the thread also passes centrally 



50 THE BINARY STARS 

through the companion star. It is most convenient to manipu- 
late the box screw with the left hand and the pinion with the 
right. 

The tangent screw giving a slow motion in position angle 
is never used; in fact, it has been removed from the micrometer. 
When the seeing is good, the star images round, small, and 
steady, it is easy to hold both images on the thread until the 
eye is assured of their precise bisection. Under less favorable 
conditions a rapid to and fro motion of the box screw places 
the stars alternately on either side of the thread while the 
pinion is being rotated backward and forward until the eye 
is satisfied of the parallelism of the thread to the line joining the 
centers of the star images. 

Ordinarily four independent settings for position angle are 
made, the circle being read, not by the vernier but by an index, 
directly to half degrees in the case of the 12-in. micrometer, 
to quarter degrees in the case of the 36-in., and by estimation 
to the one-fifth of a division, i.e., to 0?1 and 0°05, respectively. 
To insure the independence of the readings, the micrometer is 
rotated backward and forward through an arc of 60° to 80° 
after each setting. The eye is, of course, removed from the 
eyepiece, and the box is turned directly with the hands, without 
the use of the rotation pinions. 

The circle is next set to a reading 90° greater (or less) than 
the mean of the readings for position angle and the distance 
is measured by bisecting one star with the fixed thread, the 
other with the micrometer thread. It is most convenient to 
turn the micrometer screw with the right hand, the box screw 
with the left. Then interchange the threads, placing the 
micrometer thread on the first star, the fixed thread on the 
other. The difference between the two readings of the microm- 
eter screw-head gives the double distance, i.e., twice the 
angular separation, and eliminates the zero or coincidence 
reading. Three measures of the double distance are generally 
made. The milled head of the screw, which is divided to 
hundredths of a revolution, is read to the 1/1,000 revolution by 
estimation. Care is always taken to run the micrometer thread 
back several seconds of arc after each setting and to make 
the final turn of the screw at each bisection forward or against 
the springs. The bisection of the star by the fixed thread should 
be made anew at each setting with the micrometer screw, because, 



OBSERVING METHODS, VISUAL BINARY STARS 51 

under even the best conditions, it cannot be assumed that the 
star images will remain motionless during the time of observation. 
Any ordinary notebook will answer as a record book. At 
the Lick Observatory, we have found convenient a book 7 by 
8% in. containing 150 pages of horizontally ruled, sized paper 
suitable for ink as well as pencil marks. The observing record 
is made with pencil, the reductions in ink. No printed forms 
are necessary or even convenient. A sample entry taken from 
my observing book shows the form of record adopted, and also 
the very simple reductions: 









36" Sat. Jan. 27, 1917 


80Tauri = 2 554 




Parallel = 10?25 


128?75re/0 


9 ± 


49.401 9.581 


4^3 


129.70 Am 


= 3 


9.400 9.578 


1,000 


129.30 




9.403 9.580 


2to2 + 


130.40 






Well separated with 








49.401 9.580 
9.401 


520-power 


129.54 




100.25 
















2)0.179 





29.°3 = O 0.089# = 88 = p 

Two or three such entries are ordinarily made to the page. The 
column at the left records the four settings for position angle; 
the mean is taken and the reading of the circle for parallel 
plus 90° is subtracted to obtain the position angle. Whether 
this value is the correct one or whether 180° is to be added to 
it is decided by the note made of the quadrant while observing 
— nf in the present case. When recording the quadrant, which 
is done after the position angle settings have been entered, 
I record also an estimate of the distance and of the difference of 
magnitude of the components, or, when the companion is very 
faint, a direct estimate of its magnitude. At this time, too, I 
record, at the right, the date, the sidereal time to the tenth of 
an hour, the power of the eyepiece used, an estimate of the seeing 
on a scale on which 5 stands for perfect conditions and any 
observing notes» Measures of distance are then made and 
recorded. Here the reduction consists in taking half the differ- 
ence of the two means and multiplying the result by the value 
of one revolution of the micrometer screw (in this instance 
9 -'907). 



52 THE BINARY STARS 

The results are transferee! to a "ledger," or, preferably, to 
the cards of a card catalogue, the date being recorded as a decimal 
of the year. The ledger entry for the above observation is : 

80 Tauri = 2554. 
1917.075 29°3 0'/88, Am = 3, 4^3, 1000, 2 to 2 + , bk. 87,147 

the last item being the number and page of the observing book. 

Practically all observers agree in the method of measuring 
the angular distance, but many prefer a somewhat different 
procedure for determining the position angle. They bring the 
two threads fairly close together — to a separation twice or 
three times the diameter of the primary's apparent disk — 
and then, placing the two stars between the threads, turn the 
micrometer until the line joining the stars appears to be parallel 
to the threads. I have found that I can secure equally satis- 
factory measures by this method when the two stars are well 
separated and of nearly equal magnitude, but not when the 
angular distance is small or when the stars differ much in bright- 
ness. While it may be a matter of personal adaptation I am 
inclined to think that measures made in this manner are more 
likely to be affected by systematic errors than those made by 
the method first described.* 

Whatever method is adopted, it is of the first importance 
that the head of the observer be so held that the line between 
his eyes is parallel or perpendicular to the line joining the two 
stars. I can make the bisections with more assurance when 
the line between the eyes is parallel to the one joining the two 
stars, and hold my head accordingly unless the line is inclined 
more than 45° to the horizon. Some observers prefer the 
perpendicular position. When the line joining the stars makes 
an angle of approximately 45° to the horizon, it is well to make 
settings in both positions of the eyes. In pairs with components 
of unequal magnitude, a systematic difference between the two 
sets of readings may be expected. 

There are some other precautions that must be taken to secure 
satisfactory results. The star images as well as the threads 
must be brought sharply into focus; the images must be sym- 

* It should be noted that one or two good observers determine the position 
angle by setting the threads as nearly as possible perpendicular to the line 
joining the two stars. This practice is not recommended. 



OBSERVING METHODS, VISUAL BINARY STARS 53 

metrically placed with respect to the optical axis; and the threads 
must be uniformly illuminated on either side. In modern 
micrometers the ulumination is usually provided by a small 
incandescent lamp placed in such a position that a small mirror 
can throw the light through a narrow opening in one end of the 
micrometer box. This mirror can be rotated through 90° thus 
permitting a variation in the intensity of the light from full 
illumination to zero. Suitable reflectors placed within the 
micrometer box, at the opposite end, insure equality in the illumi- 
nation on both sides of the threads. Glass slides or color filters 
can also be placed in front of the opening admitting the light 
in order to vary its intensity or its color as may be desired. 
The earlier double star observers frequently illuminated the 
field of view instead of the threads and an occasional observer 
still advocates this practice, but the great majority, I think, 
are agreed that this is a less satisfactory arrangement. 

It is hardly necessary to say that the micrometer threads 
must be stretched to a tension sufficient to keep them perfectly 
straight, even when the atmosphere is very moist, and that they 
must be free from dust or other irregularities and accurately 
parallel. A cocoon of spider thread should be obtained from 
an instrument maker and kept on hand with the necessary 
adjusting tools and the micrometer threads replaced as often 
as they become unsatisfactory. A little practice will enable 
the observer to set a thread in position in a very short space 
of time; in fact, from Burnham's days to the present time, a 
new thread has frequently been set into the 36-in. micrometer 
during the night and observing resumed within an hour. 

The most important precaution to be taken in double star 
observing is quite independent of instrumental adjustments. 
It is to make measures only on nights when the observing condi- 
tions are good. Measures made under poor observing condi- 
tions are at best of little value, and at worst are a positive 
nuisance to the student of double star motions. They annoy 
or mislead him in his preliminary investigations and are prac- 
tically rejected in his later work. I make this statement with 
all possible emphasis. 

It is of almost equal consequence to select stars suited to 
the power of the telescope employed. This, however, is to a 
considerable extent a matter involving the personal equation. 
A Dawes, a Dembowski, or a Burnham can measure with 



54 



THE BINARY STARS 



comparatively small apertures stars that other observers find 
difficult with much larger telescopes. 

MAGNITUDE ESTIMATES 

It is well known that the magnitudes assigned to the com- 
ponents of the same double star by different observers fre- 
quently show a range that is excessively large. Whatever excuse 
there may have been for this in earlier days, there is certainly 
little at the present time when the magnitudes of all of the 
brighter stars are given in the photometric catalogues and 
those of all stars to at least 9.5 magnitude in the various 
Durchmusterungen. It is certainly advisable to take the com- 
bined magnitude of the two components (or the magnitude of 
the brighter star, if the companion is very faint) from these 
sources instead of making entirely independent estimates. 
The difference of magnitude is then the only quantity the double 
star observer need estimate. If this difference is not too 
great it can be estimated with comparative accuracy; if one 
component is very faint, a direct estimate of its brightness 
may be based upon the limiting magnitude visible in the tele- 
scope used, care being taken to allow for the effect of the bright 
companion which will always make the faint star appear fainter 
than it really is. 

To derive the brightness of each component when the com- 
bined magnitude and the difference of magnitude are known, 
we have the relations, A = C + x, B = A + d, in which A and 
B are the magnitudes of the brighter and fainter component, 
respectively, C the combined magnitude, and d the estimated 
difference of magnitude, while x is given by the equation 



x = 



) 



0.4 



We may tabulate x for different values of d as in the follow- 
ing table which is abbreviated from the one in Innes' Reference 
Catalogue: 



d 


X 


d 


X 


0.0 


0.75 


1.5 


0.25 


0.25 


0.6 


2.0 


0.15 


0.5 


0.5 


2.5 


0.1 


0.75 


0.4 


3.0 


0.05 


1.0 


0.3 


4.0 


0.0 



OBSERVING METHODS, VISUAL BINARY STARS 55 

To illustrate the use of the table let d, the observed difference 
in brightness, be 0.7m (it is desirable to estimate the difference 
to the nearest tenth of a magnitude), and let the photometric 
magnitude, C, be 7.0. Then, from the table, x = 0.4, and the 
magnitudes of A and B are 7.4 and 8.1 (to the nearest even tenth). 
Conversely, we may find C from A and B. 

THE OBSERVING PROGRAM 

It has happened in the past that certain well-known double 
stars have been measured and remeasured beyond all reasonable 
need, while other systems of equal importance have been almost 
entirely neglected. The general catalogues described in the 
preceding chapter make it comparatively easy for observers 
to avoid such mistakes hereafter. In the light of the knowledge 
these catalogues give of past observations and of the motions 
in the various systems, the observer who wishes his work to be of 
the greatest possible value will select stars which are suited to 
his telescope and which are in need of measurement at a given 
epoch either because of scarcity of earlier measures or because 
the companion is at a critical point in its orbit. 

It. has often been said that a careful set of measures of any 
pair of stars made at any time is valuable. Granting this to 
be so, it is certain that its value is greatly enhanced if it is 
made to contribute to the advancement of a program having 
a definite end in view. If the aim is to increase the number 
of known orbits as rapidly as possible, attention should be 
centered upon the closer pairs, particularly those under 0"5 
and those which have already been observed over considerable 
arcs of their orbits. Though I have felt myself obliged to devote 
my observing time in recent years to the remeasurement of as 
many as possible of the pairs of my own discovery, I am none 
the less convinced that it is, in general, wise for an observer 
possessing the necessary telescopic equipment to devote his 
energy largely to the measurement of a limited number of 
rapidly moving systems, repeating the measures every year, 
or every two or five years, as may be required by the rapidity 
of the orbital motion, for a long series of years. Such a series 
can be investigated for systematic as well as accidental errors 
of measure far more effectively than an equal number of meas- 
ures scattered over a much larger program, and will add more 
to our knowledge of the orbits of the binary systems. The 



56 THE BINARY STARS 

wider pairs, and particularly those in the older catalogues, 
now need comparatively little attention, so far as orbital motion 
is concerned. Even moderately close pairs, with distance 
from 1" to 5", need, in general, to be measured but once in 
every 10 or 20 years. Useful programs, however, may be 
made from wider pairs for the detection of proper motions, 
or for the determination of the relative masses in binary systems 
by means of measures connecting one of the components with 
one or more distant independent stars. Photographic measures 
of these wider pairs are, in general, more accurate than visual 
ones. 

It is hardly necessary to add that an hour in the dome on 
a good night is more valuable than half a dozen hours at the 
desk in daylight. Everything possible should therefore be 
done to prevent loss of observing time. In this connection I 
have found charts based on the Durchmusterung invaluable 
for quick identification of stars. 

THE RESOLVING POWER OF A TELESCOPE 

It has been shown that the diffraction pattern of the image 
of a point source of light, like a star, formed by a lens "is a 
disk surrounded by bright rings, which are separated by cir- 
cles at which the intensity vanishes."* 

Schuster gives the formula 

P=rnf± (7) 

in which p is the radius of a circle of zero intensity (dark ring), 
D the diameter of the lens, / its local length, \ the wave length 
of the light from the point source, and m a coefficient that 
must be calculated for each ring. For the first dark ring it is 
1.220, and the values for the successive rings increase by very 
nearly one unit. Nearly all of the light (0.839) is in the cen- 
tral disk, and the intensity of the bright diffraction rings falls 
off very rapidly. Now it is generally agreed that the mini- 
mum distance at which a double star can be distinctly seen 
as two separate stars is reached when the central disk of the 
image of the companion star falls upon the first dark ring of 
the image of the primary, and the radius of this ring, expressed 
in seconds of arc, is therefore frequently called the limit of the 

* Schuster, Theory of Optics, p. 130, 1904. 



OBSERVING METHODS, VISUAL BINARY STARS 57 

telescope's resolving power. If we adopt for X the wave length 
5,500 A, the expression for p in angular measure becomes 

A - 5r45 (to 

P - ~p- (8) 

from which the resolving power of a telescope of aperture D 
(in inches) may be obtained. For the 36-in. Lick refractor, the 
formula gives 0T15, for the 12-in., 0:'45. 

It will be observed that the resolving power as thus derived 
rests partly upon a theoretical and partly upon an empirical 
basis. When the central disk of each star image of a pair falls 
upon the first dark ring of the other image, the intensity curve 
of the combined image will show two maxima separated by a 
distinct minimum. When the disks fall closer together, this 
minimum disappears, the image becomes merely elongated, 
perhaps with slight notches to mark the position of the dis- 
appearing minimum. The pair is now no longer "resolved," 
according to the definition given, but to the experienced observer 
its character may still be unmistakable. For example, in the 
Lick Observatory double star survey, Hussey and I have found 
with the 36-in. at least five double stars with measured distances 
of O'-'ll or less, the minimum for each observer being 0"09; 
and we have found many pairs with the 12-in. telescope whose 
distances, measured afterward with the 36-in., range from 0"20 
to 0"25. In all these cases the magnitudes were, of course, 
nearly equal. 

Lewis* published a very interesting table of the most difficult 
double stars measured and discovered by various observers 
using telescopes ranging in aperture from 4 to 36 in. He tabu- 
lated in separate columns the values for the bright and faint 
pairs of nearly equal magnitude, and for the bright and faint 
pairs of unequal magnitude, each value representing the mean 
of about five of the closest pairs for a given observer and telescope. 
A final column gave the theoretical resolving power derived, not 
from the equation given above, but from Dawes' well-known 
empirical formula — resolving power equal 4'- 56 divided by the 
aperture in inches (a) — which assumes the two stars to be of 
about the sixth magnitude. Lewis found that, in general, this 
formula gave values which were too small even for the bright 

* The Observatory, 37, 378, 1914. 



58 THE BINARY STARS 

equal pairs, and he suggested the following as representing more 
precisely the results of observation : 

4'.'8 
Equal bright pairs > mean magnitudes 5.7 and 6.4 

8'-' 5 
Equal faint pairs — — > mean magnitudes 8.5 and 9.1 

16'.'5 
Unequal pairs — — > mean magnitudes 6.2 and 9.5 

36'-'0 
Very unequal pairs — — > mean magnitudes 4.7 and 10.4 

Lewis was careful to state that his table did not necessarily 
represent the minimum limits that may be reached with a 
given telescope under the best conditions, and I have just 
shown that it does not represent the limits actually reached 
at the Lick Observatory. Taking from each of the three lists 
of new double stars 1,026 to 1,274, Hu 1 to Hu 1,327, and 
A 1 to A 3104 "about five" of the closest bright, and closest 
faint, equal pairs discovered by each of the three observers, 
Burnham, Hussey, and Aitken — 29 pairs in all — I find the 
following formula for the 36-in. telescope: 

4 '.'3 
Equal bright pairs —^—t mean magnitudes 6.9 and 7.1 

6'/l 
Equal faint pairs -^— > mean magnitudes 8.8 and 9.0 

The most interesting point about these formulas is that 
they show much less difference between the values for faint 
and bright pairs than Lewis' do. 

While it is a matter of decided interest to compare the limits 
actually attained with a given telescope with the theoretical 
resolving power, an observer, in making out his working pro- 
gram for double star measurement, will do well to select pairs 
that run considerably above such limiting distances. My 
deliberate judgment is that, under average good observing con- 
ditions, the angular separation of the pairs measured should 
be nearly double the theoretical limit. Observers with the 
most powerful telescopes, however, are confronted with the 
fact that if they do not measure the very closest known pairs 
these must go unmeasured.* 

* Some of them may, however, be measured with the interferometer. 
The method is described on p. 67. 



OBSERVING METHODS, VISUAL BINARY STARS 59 

EYEPIECES 

The power of the eyepiece to be used is a matter of practical 
importance, but one for which it is not easy to lay down spe- 
cific rules. The general principle is — use the highest power the 
seeing will permit. When the seeing is poor, the images "danc- 
ing" or blurred, increase in the magnifying power increases 
these defects in the images and frequently more than offsets 
in this way the gain from increase in the scale. On such nights, 
if they are suitable for any work, choose wider pairs and use 
lower powers. The practical observer soon realizes that it is 
not worth while to measure close pairs except with high powers. 
With the 36-in. telescope my own practice is to use an eyepiece 
magnifying about 520 diameters for pairs with angular separation 
of 2" or more. If the distance is only 1", I prefer a power of 
1,000, and for pairs under 0"5 I use powers from 1,000 to 3,000, 
according to the angular distance and the conditions. The 
closeness and brightness of the pair and the quality of the 
definition are the factors that determine the choice. Very 
close pairs are never attempted unless powers of 1,500 or higher 
can be used to advantage. 

The simplest method of measuring the magnifying power of 
an eyepiece in conjunction with a given objective is to find 
the ratio of the diameter of the objective to that of its image 
formed by the eyepiece — the telescope being focused and directed 
to the bright daylight sky. Two fine lines ruled on a piece of 
oiled paper to open at a small angle form a convenient gage 
for measuring the diameter of the image. A very small error 
in this measure, however, produces a large error in the ratio 
and the measure should be repeated many times and the mean 
result adopted. The magnifying power of an eyepiece may, 
of course, also be measured by a dynameter if one is available. 

DIAPHRAGMS 

It is sometimes said that the quality of star images is improved 
by placing a diaphragm over the objective to cut down its 
aperture. I question this. It is certain that the experience 
of such observers as Schiaparelli and Burnham was directly 
opposed to it, and experiments made with the 12-in. and 36-in. 
telescopes offer no support for it. Indeed, it is difficult to 
understand how cutting off part of the beam of light falling 



60 THE BINARY STARS 

upon an object glass of good figure can improve the character 
of the image, unless it is assumed that the amplitude of such 
atmospheric disturbances as affect the definition is small enough 
to enter the problem. The only possible gain might be in the 
reduction of the brightness of the image when one star of a pair 
is exceptionally bright, as in Sirius; but this reduction can be 
effected more conveniently by the use of colored shade glasses 
over the eyepiece. These are occasionally of advantage. * 

A hexagonal diaphragm placed over the objective, however, 
may prove of great value in measuring stars, like Sirius or 
Procyon, which are attended by companions relatively very 
faint; but this is because such a diaphragm entirely changes 
the pattern of the diffraction image of the star, not because 
it cuts down the aperture of the telescope. The pattern is 
now a central disk from which six thin rays run; between these 
rays the field appears dark even close to the bright star, and 
a faint object there can be seen readily that would be invisible 
otherwise. Professor Barnard f used such a diaphragm to 
advantage with the 40-in. Yerkes refractor. Provision should 
be made for rotating the diaphragm through an angle of about 
60° and it will be convenient in the case of a large instrument 
to be able to do this by means of gearing attached to a rod 
running down to the eye end. 

ERRORS OF OBSERVATION 

All measures of angles or of distances are affected by errors, 
both accidental and systematic, and when, as in double star 
work, the measured quantities are very minute, these errors 
must be most carefully considered. The accidental errors 
may be reduced by careful work and by repeating the meas- 
ures a suitable number of times. Little is to be gained, in this 
respect or in any other, by making too large a number of set- 
tings upon an object on any one night; because such factors 
as the seeing, the hour angle, the observer's physiological con- 
dition, all remain nearly constant. As a rule, four settings for 
position angle and three measures of double distance are enough 

* Van den Bos, in a recent letter, argues that under poor atmospheric 
conditions, an iris diaphragm, to reduce aperture, is helpful, particularly 
in the measurement of unequal or very bright pairs. 

t A.N. 182, 13, 1909. 



OBSERVING METHODS, VISUAL BINARY STARS 61 

to make on one night, but the measures should be repeated 
on one or more additional nights. This is not only to reduce 
the accidental error of measure but to guard against outright 
mistakes in reading the circles, recording, etc. As to the number 
of nights on which a system should be measured at a given epoch, 
opinions will differ. Some observers run to excess in this matter. 
Generally, it may be said that it is time wasted to measure a 
system on more than four nights at any epoch and ordinarily the 
mean of three nights' measures, or even of two, if the pair is easy 
to measure and the measures themselves are accordant, is as 
satisfactory as the mean from a larger number. In critical cases, 
however, a larger number is sometimes desirable. 

The systematic errors of measurement are far more trouble- 
some, for they vary not only with the individual but are differ- 
ent for the same observer at different times and for different 
objects. Aside from the personality of the observer, they 
depend upon the relative magnitudes of the two components 
of a double star, the angular distance, the angle which the line 
joining the stars makes with the horizontal, and, in unequal 
pairs, upon the position of the faint star with respect to the 
bright one. Various methods have been adopted to deter- 
mine these errors or to eliminate them. 

The most elaborate investigation in this line is probably 
the one made by Otto Struve, who measured "artificial double 
stars formed by small ivory cylinders placed in holes in a black 
disk." He deduced formulae by means of which he calculated 
corrections to be applied to all his measures; but it is very 
doubtful whether these corrections really improve the results. 
I agree with Lewis when he says, "I would prefer his original 
measures— in part because the stars were so particularly arti- 
ficial." The actual conditions when observing the stars at 
night are of necessity widely different from those under which 
the test measures were made. Certainly, in the case of Otto 
Struve, the corrected angles and distances are frequently more 
at variance with the general run of all of the measures by good 
observers than the original values. The student of double 
star motions will generally find it advantageous to use the 
original uncorrected measures of every observer in his pre- 
liminary work and then to derive values for the systematic 
or personal errors of each by comparing his measures with 
the curve representing the means of all available measures. 



62 



THE BINARY STARS 



The observer, on the other hand, may profitably adopt 
observing methods designed to eliminate, in part at least, 
systematic errors. Innes' plan of measuring each pair on 
each side of the meridian is an excellent one because, in gen- 
eral, the line joining the two stars changes its angle with respect 
to the horizon in passing the meridian. In the extreme case, 
if the smaller star is above the primary when the pair is east 
of the meridian, it will be below when west of the meridian. 
When Innes' two measures made in this way were not sufficiently 
accordant, he repeated them on two additional nights, one night 
in each position of the instrument. 

In 1908, MM. Salet and Bosler* published the results of an 
investigation of the systematic errors in measures of position 
angle in which they made use of a small total reflecting prism 
mounted between the eyepiece and the observer's eye and 
capable of being rotated in such manner as to invert the field 
of view. Theoretically, half the sum of the measures made 
without and with the prism should represent the angle freed 
from errors depending upon the inclination of the images to 
the horizon. In fact, Salet and Bosler found that, whereas 
their measures without the prism and those made with it both 
showed a personal equation varying in amount with the star, 
the means of the two sets were remarkably free from person- 
ality. Here, for example, are their measures of y Leonis: 



Observer 


Date 


Without prism 


With prism 


Mean 


Salet 
Bosler 

(S-B) 


1907.19 
1907.23 


119-04 
116.80 

+2.24 


113-50 
116.07 
-2.57 


116-27 
116.44 
-0.17 



Hermann Struve and J. Voute later published measures made 
in this manner and each concluded that the results were far better 
than his measures made entirely without the use of the prism. 
In one of his papers t Voute states that "it is principally in 
observing in the perpendicular ( :) position that the observations 
show a pronounced systematic error," while "the parallel (. .) 
observations are in general free from systematic errors." 

* Bull. Astronomique 26, 18, 1908. 
t Union Obs. Circ. 27, 1915. 



OBSERVING METHODS, VISUAL BINARY STARS 



63 



Dawes* long ago pointed out that in "rather close double 
stars," the measures of distance "will almost inevitably be 
considerably too large" unless the observer has taken into 
account the change made in the apparent form of the star 
disk when a thread of the micrometer is placed over it. This 
change is in the nature of a swelling out of the disk on each 
side of the thread, producing an approximately elliptical disk. 
When two images are nearly in contact and the threads are 
placed over them, this swelling obliterates the interval between 
the disks and the threads are therefore set too far apart. The 
effect disappears when the disks are well separated. 

In my investigations of double star orbits I have frequently 
noticed that distance measures of a given system made with 
small apertures are greater than those made with large telescopes 
even when made by the same observer, provided the system is a 
close one as viewed in the smaller instrument. I have found such 
a systematic difference in the distances in stars which I have 
measured with the 12-in. and with the 36-in. telescope, and 
Schlesingerf has also called attention to this difference, giving a 
table derived from my measures as printed in Vol. XII of the 
Publications of the Lick Observatory. This table is here reproduced 
with a column of differences added : 

Measured Separations 



Number of stars 


With the 12-in. 


With the 36-in. 


Difference 


20 


0''52 


0-'42 


H-0'.'IO 


25 


0.62 


0.54 


+0.08 


20 


0.71 


0.64 


+0.07 


24 


0.81 


0.79 


+0.02 


24 


1.07 


1.03 


+0.04 


21 


1.38 


1.39 


-0.01 


26 


2.13 


2.10 


+0.03 


18 


4.49 


4.53 


-0.04 



The systematic difference is clearly shown in all the pairs 
having a separation less than twice the resolving power (0"42) 
of the 12-in. telescope; in the wider pairs it is negligibly small. 

Occasionally an observer's work shows systematic differences 
of precisely the opposite sign. Thus Schlesinger (loc. cit.) 

* Mem. R. A. S., 35, 153, 1867. 
t Science, N. S., 44, 573, 1916. 



64 THE BINARY STARS 

in analyzing the measures by Fox in the Annals of the Dearborn 
Observatory, Vol. I, (1915) finds that the distances are measured 
smaller with the 12-in. than with the 182^-in. or with the 40-in., 
"the differences being largest for small separations and becoming 
negligibly small for separations in the neighborhood of 5"." 
The personal equation revealed in such comparisons as these 
must obviously be taken into account in orbit computations. 

PHOTOGRAPHIC MEASURES 

The first double star to be discovered visually was also the 
first for which measurable photographic images were obtained. 
G. P. Bond photographed f Ursae Majoris, angular separation 
14'.'2, on a collodion plate in 1857, giving an 8-sec. exposure. 
Pickering and Gould in America, M. Henry in France, and the 
Greenwich observers in England, among others, followed up 
this early success and secured results of value for a number of 
pairs, a few of them as close as 1". More recently, extensive 
programs have been carried out at several observatories and it 
has become evident that photographic measures, made under 
proper conditions, are of the same order of accuracy as visual 
ones for pairs with angular distances of from 1" to 2", if the 
components are not very unequal in magnitude, and that they 
exceed visual measures in accuracy for wider pairs. A photo- 
graphic observation of a double star, moreover^ has the great 
advantage, as Hertzsprung has pointed out, of being a "per- 
manent document " which can be reexamined as often as may be 
desirable. 

Hertzsprung, working at Potsdam in the years 1914 to 1919, 
made a thorough investigation of the possibilities of the photo- 
graphic method as applied to the measurement of double 
stars, including a study of the sources of accidental and system- 
atic errors, and of the procedure necessary to eliminate such 
errors or, failing this, to reduce to a minimum their effects upon 
the measures. 

It appears that the chief sources of error are (1) the difference 
in magnitude of the components of a pair, (2) refraction, and 
more particularly, the change of refraction with the color of a 
star, and (3) the Eberhard effect, or the effect upon a photo- 
graphic star image produced by the close proximity of another 
star image. There are, of course, also the accidental errors of 
measurement, and the errors arising from incorrect scale value 



OBSERVING METHODS, VISUAL BINARY STARS 65 

and imperfect orientation, errors that are comparable to those 
in visual observations from corresponding sources. 

1. The Magnitude Equation. — The images of the two com- 
ponents of a double star are so nearly in contact on the photo- 
graphic plate that the methods of eliminating the error arising 
from difference in magnitude in use in parallax determinations 
and other precise photographic measurements are not applicable. 
A rotating sector, for example, would, for all but the relatively 
wide pairs, cut down the light from both components in sub- 
stantially the same degree. 

The method that has given the best results up to the present 
time is the one adopted by Hertzsprung and others of covering 
the objective of the telescope with a coarse grating made of rods 
of uniform thickness, uniformly spaced. Such a grating will 
give, in addition to the principal image of a star, spectra sym- 
metrically placed on either side of it, and if it is made so coarse 
that the first-order spectra are just clearly separated from the 
principal image, they, too, will have sensibly stellar images. 
Further, by proper choice of rod thickness and free spacing the 
difference in magnitude between the principal image and its 
first-order spectra may be made to vary from one to four or more 
magnitudes. It is well to have a set of six or more such gratings 
so calculated that the magnitude difference may be varied by 
steps of approximately a magnitude. It will then be possible 
in practically all cases to bring the brightness of the first-order 
spectra of the primary within half a magnitude of that of the 
principal image of the companion. A difference of half a 
magnitude will have little or no effect upon the measures. The 
gratings, it is to be noted, will reduce the brightness of the 
principal images of both components by an amount depending 
upon the grating constant, and the exposure time will also vary 
with the grating used. 

The practical difficulties in using this method with telescopes of 
large aperture arise from the expense involved in constructing 
good gratings of large size and from the inconvenience of inter- 
changing them in the course of a night's observing. 

2. Refraction and Star Color. — On account of atmospheric 
dispersion, the refraction varies with the effective wave length 
of a star's light. A systematic error of measurement may, 
therefore, result when the colors of the two components of a 
double star differ appreciably. To eliminate it, take the photo- 



66 THE BINARY STARS 

graphs in light of a restricted range in wave length, by using 
appropriate niters and, if necessary, special plates. The particu- 
lar niters and plates required will vary with the telescope used. 
It is desirable also, to avoid taking photographs at great zenith 
distances. 

3. The Photographic Star Image. — Eberhard, many years 
ago, showed that the work done by a developer in blackening 
any small area on a photographic plate depended upon how 
much more work it had to do in the immediate vicinity. The 
density of a star image, for a given exposure and development, 
therefore, depends not only upon the brightness of the star but in 
part also upon the presence of other star images in its immediate 
proximity. In the case of a double star the outer edge of the 
image of each component will be denser than its inner edge — 
the edge nearest the other component. The effective centers 
of the two images are therefore displaced in opposite directions 
and the apparent or measured distance is larger than the true 
distance between them. But Hertzsprung also found evidence 
of a displacement in the opposite sense in the case of some photo- 
graphs of very close pairs taken under relatively poor conditions. 
Apparently, however, both effects appear only in pairs in which 
the images of the two components are nearly or quite in contact. 
For pairs with clearly separated images, the error from this source 
is so small that it may be neglected. 

In addition to the precautions to be taken with respect to the 
sources of systematic error just described, it is, of course, neces- 
sary to put the plate accurately in focus, and to determine the 
parallel carefully. To provide means for determining the 
parallel a bright star in the immediate neighborhood may be 
allowed to trail on the plate, or images of the double star (with 
the plate displaced slightly in declination) may be taken both 
near the preceding and near the following edge of the plate, 
the clock being stopped between the two exposures. The 
exposure time will depend upon the star and the telescope. 
It is desirable to have distinct images but not denser than 
necessary for easy measurement. 

Finally, the importance of photographing only on calm nights, 
and of having an accurate driving mechanism for the telescope 
is to be particularly emphasized. 

The plates thus secured should be measured in rectangular 
coordinates and it will be advantageous to use an engine in which 



OBSERVING METHODS, VISUAL BINARY STARS 67 

the plate, rather than the eyepiece, is moved by the micrometer 
screw. In this case the value of a revolution of the screw, in 
seconds of arc, is independent of the power of the eyepiece, and 
the plate may be measured both film side up and through the 
glass. It is hardly necessary to add that the scale value of the 
plates must be determined accurately and that the plates must be 
oriented with great care. Since the first-order spectra are 
symmetrically placed with respect to the center of the principal 
image, the mean of their positions may be adopted as the position 
of the primary. 

It has already been noted that in recent years lists of new 
double stars detected on parallax plates and lists of others found 
in the examination of the astrographic star catalogue for different 
zones have been published by Scheiner, Stein, Barton, Olivier, 
and others. The pairs in the astrographic lists are all relatively 
wide and for the most part faint, and the measures of position 
angles and distances are not of a high order of accuracy. The 
lists from parallax plates also consist of faint pairs, but some of 
them have angular distance not greatly in excess of a second 
of arc. The measures of these pairs are probably quite as 
accurate as visual measures would be. 

INTERFEROMETER MEASURES 

The advantages of the photographic method are so great that 
it will undoubtedly come into ever more general use in the 
measurement of all double stars that can be resolved on the 
photographic plate. The closer pairs, and particularly those 
whose components differ greatly in magnitude, must be left to 
the visual observer working with the filar micrometer, with or 
without the reversing prisms, or instruments that are improve- 
ments upon it. Hargreaves, for example, has recently designed 
a comparison image micrometer* that promises to give more 
accurate results, for the angular distance, at least, than the 
filar micrometer. 

The lower limit for accurate micrometric measurements of 
angular separation with existing telescopes is about 0'-'13, 
though under the best conditions fairly good estimates may be 
made for pairs as close as O'-'IO. The only instrument that 
promises good results for still closer pairs is the interferometer. 

* See footnote, p. 41. 



68 THE BINARY STARS 

Michelson, more than 40 years ago, showed that such extremely 
small angles as the diameter of a small satellite or the distance 
between the components of a double star could be measured 
"by observing the interference fringes produced at the focus 
of a telescope when only two portions of the objective, located 
on the same diameter, are used." He demonstrated that "as 
the distance apart of the apertures is increased the visibility of 
the fringes reaches a minimum for a distance equal to 1.22X/a, 
or 0.5X/a, for a disk or double star, respectively, where X is the 
effective wave length and a is the desired angle." He applied 
the method to the measurement of the diameters of the satellites 
of Jupiter, with the 12-in. refractor of the Lick Observatory 
in 1891. Four years later,* Schwarzschild applied the principle 
to the measurement of double stars, using a specially designed 
interferometer placed over the objective of the 10-in. refractor 
at Munich. He measured a number of pairs with angular 
distances ranging from 0"9 to 3"7. No one else seems to have 
made interferometer measures of double stars until Anderson,f 
early in 1920, using an interferometer of his own design, attached 
to the 100-in. reflector at Mount Wilson, measured the position 
angle and distance of Capella, with an accuracy far exceeding 
the accuracy of the best micrometer measures of a close double 
star. Merrill,! using the same instrument, secured additional 
measures of Capella later in the same year and in 1921, and also 
measured k Ursae Majoris = A1585, which he and Anderson 
had discovered as a double, by the use of the interferometer, before 
they knew of my earlier micrometric measures. My measures 
had shown a rapid decrease in angular distance, the value in 
1919.29 being 0'-'15. Their measures, in 1921, gave 0'-'08, 
with a change in the position angle that was consistent with the 
law of areas. Attempts to find other double stars by examination 
with the interferometer gave negative or doubtful results, and 
the work had presently to be discontinued. 

Experiments made at the Lick and at the Yerkes Observatories 
did not prove satisfactory. At Catania, M. Maggini,§ in 1922, 
applied an interferometer to the 12-in. equatorial and in the 
years immediately following made a considerable number of 

* A.N. 139, 353, 1896. 
t Contr. ML Wilson Obs. 9, 225, 1920. 
t Contr. ML Wilson Obs. 11, 203, 1922. 
§ Pub. R. Oss. Catania, 1925. 



OBSERVING METHODS, VISUAL BINARY STARS 69 

measures of double stars with angular distances up to 0"25 and 
even greater. These pairs were all within the measuring range 
of the micrometer attached to larger telescopes, and a com- 
parison of actual measures could thus be made. In the majority 
of cases, Maggini's results were in fair to good accord with those 
made with the micrometer, but in a number of cases, the dis- 
agreement was far beyond the error of micrometric measurement, 
and his values were entirely out of accord with the earlier (and 
later) micrometric results, though the pairs in question were 
as well suited to measurement by both methods as the others 
on his list. It is clear that further investigation is required 
before the interferometer can be accepted as a standard instru- 
ment for double star observation. We must know how to 
distinguish between genuine fringes and fringe disappearance 
and spurious effects of any kind whatever; and we must determine 
the limits of magnitude and angular separation within which the 
interferometer can be applied successfully with a telescope of 
given aperture and focal length. It may be found that the 
number of pairs that can be measured with the interferometer 
is small, but, by way of compensation, these will, in general, 
be pairs that cannot be measured successfully by any other 
method unless the inclination of the orbit plane is high enough 
to permit measures of the relative radial velocities of the two 
components. It is greatly to be desired that experiments with 
the interferometer be continued. 

[The reader is referred to the second volume of the series Stars and Stellar 
Systems edited by Gerard P. Kuiper and Barbara M. Middlehurst. Volume 
II, Astronomical Techniques, was edited by William A. Hiltner. Chapter 19, 
"Techniques for Visual Measurements," by P. Muller, is a very fine discussion 
of instruments and their uses as related to visual binary stars. Chapter 22, 
by W. H. van den Bos, "Orbit Determinations of Visual Binaries"; chapter 
23, "Spectroscopic Binaries," by R. M. Petrie; and chapter 24, "Eclipsing 
Binaries," by John B. Irwin, are excellent treatments of their subjects. 

Volume III of this same series, to be published sopn, will also contain much 
relevant material. — J.T.K.] 



CHAPTER IV 

THE ORBIT OF A VISUAL BINARY STAR 

We have seen that Sir William Herschel, by his analysis of 
the observed motion in Castor and other double stars, demon- 
strated that these systems are "real binary combinations of 
two stars, intimately held together by the bonds of mutual 
attraction." Later observation has shown that the apparent 
motion in such systems is on the arc of an ellipse and that the 
radius vector drawn from the primary star to its companion 
sweeps over areas which are proportional to the times. It has 
therefore been assumed from the beginning that the attractive 
force in the binary star systems is identical with the force of 
gravitation in our solar system, as expressed by Newton's law, 
and the orbit theories which we are to investigate in the present 
chapter are all based upon this assumption. Before taking 
up the discussion of these theories it is pertinent to inquire 
whether the fundamental assumption is justified. 

It is supported by all of the available evidence, but rigorous 
mathematical proof of its validity is difficult because the 
motion which we observe in a stellar system is not the true 
motion but its projection upon a plane perpendicular to the 
line of sight. The apparent orbit is therefore, in general, not 
identical with the true orbit and the principal star may lie at 
any point within the ellipse described by the companion and 
not necessarily at either the focus or the center. Hence, in 
Leuschner's words, "mathematical difficulties are encountered 
in establishing a law of force which is independent of the angle 
0, the orientation." In the article quoted, Leuschner, after 
pointing out that "Newton did not prove the universality of 
the law of gravitation, but by a happy stroke of genius gener- 
alized a fact which he had found to be true in the case of the 
mutual attraction of the Moon and the Earth," proceeds to 
show that the law does hold throughout the solar system, the 
question of orientation not entering. He then says that, in 
binary systems, "when the law is arbitrarily assumed to be 

70 



THE ORBIT OF A VISUAL BINARY STAR 71 

independent of the orientation, as was found to be the case in 
the solar system, two possibilities arise, namely, either that 
the force is in direct proportion to the distance r between the 
two stars or that the Newtonian law applies. It can be shown, 
however, that when, in the case of an elliptic orbit, the force 
is proportional to r, the primary star must be in the center of 
the ellipse. As this has never been found to be the case, the 
only alternative is the Newtonian law." 

It should be clearly understood that the difficulty in demon- 
strating the universality of the law of gravitation here pointed 
out is purely mathematical. No physical reason has ever been 
advanced for a dependence of an attracting central force upon 
the orientation, and until such dependence has been proved 
we may safely proceed with our investigation of binary star 
orbits under the action of the law of gravitation. 

Until the relative masses of the two components are known 
it is impossible to determine the position of the center of gravity 
of the system and we are therefore unable to compute the 
orbits described by the two stars about that center. What 
our measures give us is the apparent orbit of one star, the 
companion, described about the other, the primary, which is 
assumed to remain stationary at the focus. It is clear that 
this relative orbit differs from the actual orbits of the two com- 
ponents only in its scale. 

The problem of deriving such an orbit from the micrometer 
measures of position angle and distance was first solved by 
Savary,* in 1827, but Encke f quickly followed with a different 
method of solution which was somewhat better adapted to 
the needs of the practical astronomer, and Sir John HerschelJ 
communicated a third method to the Royal Astronomical 
Society in 1832. Since then the contributions to the subject 
have been many. Some consist of entirely new methods of 
attack, others of modifications of those already proposed. 
Among the more notable investigators are Villarceau, Madler, 
Klinkerfues, Thiele, Kowalsky, Glasenapp, Seeliger, Zwiers, 
Howard, Schwarzschild, See, Russell, Innes, and van den Bos. 

The methods of Savary and Encke utilize four complete 
measures of angle and distance and, theoretically, are excellent 

* Savary, Conn, des Temps, 1830. 

t Encke, Berlin Jahrbuch, 1832. 

| Herschel, Memoirs R.A.S., 6, 171, 1833. 



72 THE BINARY STARS 

solutions of the problem; Herschel's method is designed to utilize 
all the available data, so far as he considered them reliable. 
This idea has commended itself to all later investigators. 
Herschel was, convinced, however, that the measures of dis- 
tance were far less trustworthy than those of position angle, 
and his method therefore uses the measures of distance simply 
to define the semimajor axis of the orbit; all of the other ele- 
ments depend upOn measures of position angle. At the time 
this may have been the wisest course, but the distance meas- 
ures of such early observers as W. Struve, Dawes, and Dem- 
bowski, and those of later observers working with modern 
micrometers, are entitled to nearly or quite as much weight as 
the measures of position angle and should be utilized in the 
entire orbit computation. 

Whatever method is adopted, the investigator must, of course, 
begin by assuring himself that he has data sufficient for a satis- 
factory computation. In deciding this, he should consider both 
the length of the observed arc and its form. With strongly 
marked curvature, a comparatively short arc may suffice, 
provided s the observations have a high degree of accuracy. 
Ordinarily, however, the arc should be long enough to cover 
both ends or elongation points of the apparent ellipse. 

Satisfactory data being given, the problem before the com- 
puter evidently consists of two parts: first, the determination of 
the apparent ellipse, or the constant of areal velocity, from the 
data of observation ; secondly, the derivation of the elements 
of the true orbit from the relations between an ellipse and its 
orthographic projection. 

THE APPARENT ELLIPSE 

Every complete observation of a double star supplies us with, 
three data: the time of observation, the position angle of the 
companion with respect to the primary, and the angular dis- 
tance between the two stars. It is clear, as Comstock pointed 
out many years ago, that the time of observation is known 
with far greater accuracy than either of the two coordinates 
of position. The relations between the times of observation 
and the motion in the ellipse should therefore be utilized; that 
is, the condition should be imposed that the law of areal veloc- 
ities must be satisfied as well as the condition that the points of 
observation should fall approximately upon the curve of an 



THE ORBIT OF A VISUAL BINARY STAR 73 

ellipse. Elementary as this direction is, it is one that has been 
neglected in many a computation. 

Theoretically, the first step in our computation should be the 
reduction of the measured coordinates to a common epoch by 
the application to the position angles of corrections for pre- 
cession and for the proper motion of the system. The distance 
measures need no corrections. Practically, both corrections 
are negligibly small unless the star is near the Pole, its proper 
motion unusually large, and the time covered by the observa- 
tions long. The precession correction, when required, can be 
found with sufficient accuracy from the approximate formula 

A0 = +0?0056 sin a sec 8 (t - t ) (1) 

The formula for the correction due to the proper motion of 
the system is 

Ad = -//' sin 8 (t - to) (2) 

where /z" is the proper motion in right ascension expressed in 
seconds of arc* 

When the measures of any binary star have been tabulated 
(with the above corrections, if required), they will exhibit 
discordances arising from the accidental and systematic errors of 
observation and, occasionally, from actual mistakes. If they are 
plotted, the points will not fall upon an ellipse but will be joined 
by a very irregular broken line indicating an ellipse only in 
a general way. It will be advisable to investigate the meas- 
ures for discordances before using them in the construction 
of the apparent ellipse and the simplest method is to plot 
upon coordinate paper first the position angles and then the 
distances, separately, as ordinates, against the times of obser- 
vation as abscissae, using a fairly large scale. Well-determined 
points (for example, a point resting upon several accordant 
measures by a skilled observer and supported by the preceding 
and following observations) may be indicated by heavier marks. 
Smooth free-hand curves, interpolating curves, are now to be 
drawn to represent the general run of the measures and in 
drawing these curves more consideration will naturally be 
given to the well observed points than to the others. Obser- 

* See "Note on the Effect of Proper Motion on Double Star Measures," 
by Alan Fletcher (Mon. Not. R.A.S. 92, 119, 1931), for a more complete 
discussion. 



74 THE BINARY STARS 

vations that are seriously in error will be clearly revealed and 
these should be rejected if no means of correcting them is avail- 
able. The curves will also show whether or not the measures 
as a whole are sufficiently good to make orbit computation 
desirable. 

If the amount of available material warrants it, the question 
of the systematic or personal errors of the observers should 
also be considered at this time. No reliable determination of 
such errors is possible unless (a) measures by the same observer 
under essentially the same conditions in at least four or five 
different years are at hand, and (6) unless the total number of 
measures by many different observers is sufficient to establish 
the general character of the curves beyond reasonable question. 
If the second condition is satisfied, the average of the residuals 
from the curve for a given observer may be regarded as his 
personal error and the corresponding correction may be applied 
to all of his measures. Two further points should be noted: 
First, the residuals in position angle should be reduced to arc 
by multiplying by the factor p/57.3 before the mean is taken, 
to allow for the effect of variations in the angular separation; 
second, the corrections should not be considered as constant 
over too long a period of time. The application of such correc- 
tions has the effect of reducing the size of the residuals, but the 
principal advantage to be gained from it is that it lessens the 
danger of giving undue weight to measures seriously affected by 
systematic errors of observation. 

After all corrections have been applied, the measures which 
are retained should be combined into annual means or into 
mean places at longer or shorter time intervals according to 
the requirements of the particular case. Several factors really 
enter into the question of the weights to be assigned to the 
individual observations in forming these means; for instance, 
the size of the telescope used, the observing conditions, the 
number of nights of observation, and the experience of the 
observer; but it will be wise, in general, to disregard all but 
the number of nights of observation, provided the telescope 
used is of adequate resolving power for the system in question 
and that the observer has not specifically noted some of his 
measures as uncertain. A single night's measure deserves 
small weight; mean results based upon from two to six nights' 
accordant measures may be regarded as of equal weight; means 



THE ORBIT OF A VISUAL BINARY STAR 75 

depending upon a much larger number of measures may be 
weighted higher. In general, a range in weights from one to 
three will be sufficient. 

If, in addition to the visual measures, photographic measures 
have been made at several epochs, the combination of the 
measures by the two different methods merits special considera- 
tion. When, as, for example, in the case of $ Ursae Majoris, 
many photographic measures of great accuracy are available, 
these should unquestionably be given high weight. If, on the 
other hand, only a few photographic measures have been made 
they may be weighted on the same basis as the visual observations. 

Having thus formed a series of normal places, we may find 
the apparent ellipse that best represents them either graphi- 
cally or by calculating the constants of the general equation 
of the ellipse with the origin at any point. This equation is 

ax 2 + 2hxy + by 2 + 2gx + 2fy + c = (3) 

which may be written in the form 

Ax 2 + 2Hxy + By 2 + 2Gx + 2Fy + 1 = (4) 

in which we must have A > 0, B > 0, and AB - H 2 > 0. 

If we assume the position of the primary star as origin, we 
may calculate the five constants of this equation from five 
normal places by the relations 

x = p sin d) ^ 

y = p cos 8j 

but it is advisable to make a least squares solution using all of 
the normal places. 

The great objection to this method is that it entirely disre- 
gards the time of observation. Moreover, the errors of obser- 
vation, small as they are numerically, are large in proportion 
to the quantities to be measured, a fact that makes it difficult 
to obtain a satisfactory ellipse without repeated trials. The 
graphical methods are therefore to be preferred. 

The simplest method, and one that in most cases is satis- 
factory, is to plot the positions of the companion star in polar 
coordinates, the primary star being taken as the origin. With 
the aid of an ellipsograph or by the use of two pins and a thread, 
an ellipse is drawn through the plotted points and is adjusted 
by trial until it satisfies the law of areas. This adjustment 



76 THE BINARY STARS 

must be made with the greatest precision and the curve of the ellipse 
drawn with great care, for the construction of the apparent 
ellipse is the critical part of the entire orbit determination. 
In my own practice I have found that the test for the law of 
areas can be made most rapidly by drawing radii to selected 
points which cover the entire observed arc and measuring 
the corresponding elliptic sectors with a planimeter. The 
comparison of the areal velocities derived from the different 
sectors at once indicates what corrections the ellipse requires. 
With a suitable ellipsograph a new ellipse is quickly drawn 
and the areas again measured. The process is repeated until 
a satisfactory ellipse has been obtained. 

Some investigators prefer the mode of procedure in con- 
structing the apparent ellipse first suggested by Sir John Herschel. 
An interpolating curve is drawn, in the manner described 
above, for the position angles only, using the mean or normal 
places. If the curve is carefully drawn, smoothly and without 
abrupt changes of curvature, it should give the position angle for 
any particular epoch more accurately than the measure at 
that epoch, for it rests upon all of the measures. From this 
curve read the times corresponding to, say, every 5° of angle, 
tabulate them, and take the first differences. Dividing these 
by the common angle difference will give a series of approximate 

dd 
values of dt/dd. But by the theory of elliptic motion pSr 

must be a constant and hence p = c^/-^- Therefore a series 

\ ad 

of relative values of the distance (expressed in any convenient 

unit) corresponding to every fifth degree of position angle 

can be derived from the table of angles. Now plot the points 

representing corresponding angles and relative distances; if 

the interpolating curve has been correctly drawn and read 

off they will all lie upon the arc of an ellipse. If they do not, 

draw the best possible ellipse among them and use it to correct 

the interpolating curve, repeating the process until the result 

is satisfactory. Finally, convert the relative into true distances 

by comparing those distance measures which are regarded as 

most reliable with the corresponding values in the unit adopted 

in the plot. 

There are at least two objections to this method: First, it 

does not make adequate use of the observed distances; and 



THE ORBIT OF A VISUAL BINARY STAR 77 

second, when the angle changes rapidly, as it does in many 
systems at the time of minimum apparent separation, it is 
almost impossible to draw the interpolating curve correctly. 
The former difficulty may in many cases be overcome by the 
following procedure: Read from the interpolating curve for 
position angles, normal values for every fourth year.* Regard 
each two consecutive values as the limiting radii of a circular 
sector, and as the radius of each sector adopt its mean observed 
distance, which may be derived, if desired, from an interpolating 
curve. Draw these circular sectors on coordinate paper and 
pass through them a free-hand curve, approximately the arc of 
an ellipse, giving sectors of equal area. In general, however, 
it is in my judgment most satisfactory to plot the positions of the 
companion star directly in polar coordinates, using normal 
places. From these the ellipse that best satisfies the law of 
areas must be found by the method of trial and correction. 

THE TRUE ORBIT 

After the apparent ellipse has been constructed graphically, 
or from the constants in the equation of the ellipse, it remains 
to derive the elements which define the form and size of the 
true orbit, the position of the orbit plane, the position of the 
orbit within that plane, and the position of the companion 
star in the orbit at any specified time. Three of the required 
elements have dynamic significance and are entirely independent 
of the space location of the system. These are the revolution 
period, the time of periastron passage, and the eccentricity. 
To these belongs the semimajor axis of the orbit when measured 
in linear units; measured in angular units it is, of course, also a 
function of the parallax of the system. The three remaining 
elements, the inclination of the orbit plane, the position of the 
line of nodes, and the angle between that line and the major 
axis are purely geometric and merely relate the orbit of the double 
star system to the orbit of the Earth. 

The first four elements may be defined formally as follows : Let 

P = the period of revolution expressed in mean solar years. 

T = the time of periastron passage. 

e = the eccentricity. 

a = the semiaxis major expressed in seconds of arc. 

* This will suffice for long-period systems; for systems of short period, 
readings at shorter intervals should be taken. 



78 THE BINARY STARS 

All authorities are agreed upon these definitions, but some 
confusion in the nomenclature and even in the systems used in 
defining the remaining elements has arisen from the fact that 
it is impossible to say, from the micrometer measures alone, 
on which side of the plane of projection (which is taken as the 
plane of reference) the companion star lies at a given time. 
In other words, we cannot distinguish between the ascending 
and the descending node, or between direct and retrograde 
motion in the ordinary sense. Further, in some systems the 
observed position angles increase with the times, in others they 
decrease. 

The following is the classical system in the form most con- 
venient when the requirements of the observer of radial velocities 
are considered as well as those of the observer with the microm- 
eter. Let 

12 = the position angle of that nodal point which lies between 
0° and 180°; that is, the position angle of the line of 
intersection of the orbit plane with the plane perpendicular 
to the line of sight. Call this merely "the nodal point," 
disregarding the distinction between ascending and 
descending nodes, 
w = the angle in the plane of the true orbit between the line 
of nodes and the major axis. It is to be measured from 
the nodal point to the point of periastron passage in 
the direction of the companion's motion and may have 
any value from 0° to 360°. It should be stated whether 
the position angles increase or decrease with the times. 
i = the inclination of the orbit plane; that is, the angle 
between the orbit plane and the plane at right angles to 
the line of sight. Its value lies between 0° and ±90° 
and may carry the double sign ( + ) or be left without 
sign until the indetermination has been removed by 
measures of the radial velocity.* When these are 
available, i is to be regarded as positive (+) if the orbital 
motion at the nodal point is carrying the companion 
star away from the observer; negative, if it is carrying 
the companion star toward the observer. 

* Van den Bos, however, writes i in the second quadrant in all orbits in 
which the position angle decreases with the time ("A Table of Orbits of 
Visual Binary Stars," B.A.N. 3, 149, 1926), and some later computers have 
followed this convention. 



THE ORBIT OF A VISUAL BINARY STAR 79 

The symbol » denotes the mean annual motion of the 
companion, expressed in degrees and decimals, measured 
always in the direction of motion. 

The conventions of taking ft always less than 180° and of 
counting « (for which many computers use the symbol X) always 
in the direction of the companion's motion were first suggested, 
I believe, by See, and have now been adopted generally. The 
definition of i (for which some computers write 7) is the usual 
one, also. Many computers prefer to count the mean annual 
motion in the direction of increasing position angles in all systems, 
and to consider the motion negative when the angles decrease 
with the times. 

As early as 1883 T. N. Thiele* proposed a method of computa- 
tion that would replace the geometrical elements i, a>, and ft 
and the semiaxis major a by the polar coordinates, with respect 
to the center of the apparent ellipse, of the projections on the 
plane of the apparent orbit of the two points in the true orbit 
for which the eccentric anomalies are, respectively, 0° and 90°. 
More recently Innes independently worked out a system which 
is, in all essentials, the same as Thiele's. Innes' system, as 
formulated by van den Bos, is given on a later page and the 
relations between the Thiele-Innes constants and the elements 
as defined in the preceding paragraphs are there set forth. 

When the elements are known, the apparent position angle 
and the angular distance p for the time t are derived from 
the following equations: 

360° 



M = ti(t - T) - E - e sin E( 
r = a(l — e cos E) 

tan Y 2 v = J j-jr^ tan Y 2 E 

tan (0 — ft) = ± tan (v + w) cos i 

p = r cos (v + «) sec (0 — ft) 



(6) 



} (7) 



Equations (6) are the usual ones for elliptic motion, the 
symbols M, E, and v representing respectively, the mean, 
eccentric, and true anomaly, and r the radius vector. Equa- 
tions (7) convert the v and r of the companion in the true orbit 

* A.N. 104, 245, 1883. 



80 THE BINARY STARS 

into its position angle and distance in the projected, or apparent 
orbit. Position angles are generally recorded only to the 
nearest tenth of a degree in orbit computation, hence it is 
sufficiently exact to take the value of E corresponding to a 
given value of M from. Astrand's Hiilfstafeln, which hold for 
all values of the eccentricity, or the value of v directly from 
the still more convenient Allegheny Tables,* provided the eccen- 
tricity does not exceed 0.77. If the latter tables are used, 
it is convenient to derive the value of r from the equation 

r _ «(l-s') m 

(1 + e cos v) 

instead of from the third of Eqs. (6). 

KOWALSKY'S METHOD 

From the many methods of orbit computation that have been 
formulated, I have selected for presentation here those by 
Kowalsky, by Zwiers and by Innes. All three are of very general 
application and each one has its advocates among computers. 
Several other methods are useful and the student is advised to 
examine all those for which references are given at the end of 
this chapter f 

Kowalsky's methodj is essentially analytical and derives the 
orbit elements from the constants of the general equation of 
the apparent ellipse which is the orthogonal projection of the 
true orbit, the origin of coordinates being taken at S, the position 
of the primary star. This equation takes the form 

Ax 2 + 2Hxy + By 2 + 2Gx + 2Fy + 1 = 

The values of the constants A, H, B, G and F, may be com- 
puted by the method noted on page 75, but this is open to the 
serious objection that it takes no account of the law of areas. 
It is far better to follow the procedure proposed by Glasenapp§ 
and derive the values of the five constants from measures on the 
carefully drawn apparent ellipse, as follows : 

* Pvbl. Allegheny Obs. 2, 155-190, 1912. 

t See particularly the methods of Russell (1933) and of Volet (1932). 

% First published, according to von Glasenapp, in the Proceedings of the 
Kasan Imperial University, in 1873. This volume has not been accessible 
to me. 

§ Mon. Not. R.A.S. 49, 276, 1889. 



THE ORBIT OF A VISUAL BINARY STAR 81 

In the general equation of the ellipse put y = 0; then the roots 
of the roots of the resulting equation 

Ax 2 + 2Gx + 1 = 

will be the abscissae of the points of intersection of the ellipse and 
the #-axis. Representing these roots by xi and x%, we have 

A = -L, G = _^-±^? (g) 

X1X2 2x\Xz 

Similarly, by putting x = 0, we obtain 

B = J_, F = - y -^ (9) 

2/12/2 2yij/ 2 

These four constants are thus obtained by direct measurements 
of the distances from the principal star to the intersection points 
of the ellipse with the x- and j/-axes, in which care must be taken 
to regard the algebraic signs. The fifth constant, H, is then 
derived from the equation 

H = Ax2 + B y 2 + 2Gx + 2p y + x do 

2zi/ 

Measure the coordinates of several well-distributed points on 
the apparent ellipse, so chosen as to make the product xy as large 
as possible and substitute each set of values successively in 
Eq. (10). The accordance of the resulting values of H will 
depend upon the care with which the ellipse has been drawn, 
and the mean of all should be adopted. 

The values of the coefficients A, H . . . F being known, 
we proceed as follows, adopting the analysis recently given by 
W. M. Smart:* 

Construct a sphere (Fig. 3) with the principal star S as center, 
and let SL, SM, SK be the rectangular axes to which the general 
equation of the apparent ellipse, in the plane of the great circle 
LNM (at right angles to the line of sight), is referred, SL denning 
the direction to position angle 0°. Let the great circle FNAB 
define the plane of the true orbit relative to the primary. Then 
QSN is the line of nodes, and, assuming the position angles to 
increase from L toward N, LN is the longitude of the nodal 
point £2, as denned in an earlier paragraph. Let P be the peri- 
astron point, and A the corresponding point on the sphere. 

* W. M. Smart, On the Derivation of the Elements of a Visual Binary 
Orbit by Kowalsky's Method, Mon. Not. R.A.S., 90, 534, 1930. 



82 



THE BINARY STARS 



Then NA denotes co, and ANM, the inclination, i, taken to lie 
between 0° and 90°. 

Take rectangular axes SA, SB, and SC with reference to the 
plane of the true orbit, and let (£, r], o) be the coordinates of 
the companion at any time with respect to these axes. Since the 




Fig. 3. — Diagram for Kowalsky's orbit method. 

primary S is at a focus of the true orbit, the equation of the true 
ellipse is 



(£ + aeY 



+ ?i-l 



(11) 



a 2 6 2 

where 6 2 = a 2 (l — e 2 ). 

Let (h, mi, ni), (Z 2 , m%, n 2 ), (£3, wi 3 , n 3 ) denote the direction- 
cosines of SA, SB, and SC, respectively, with reference to the 
axes SL, SM, and SK. Then, drawing great circle arcs from 
each of A, B, and C in turn to L, M, and K, we can write 

Zi = cos AL Wi = cos AM, ni = cos AK, 
l 2 = cos BL m 2 = cos BM, n 2 = cos -BiiT, 
Z 3 = cos CL, m-i = cos CM, n 3 = cos CK, 

and from the appropriate spherical triangles we derive at once 

Zi = cos ft cos co — sin ft sin co cos £} 
mi = sin ft cos co + cos ft sin co cos t> (12) 

Wi = sin co sin i ' 



THE ORBIT OF A VISUAL BINARY STAR 83 

l 2 = —cos 12 sin co — sin 12 cos co cos tl 
m,2 = —sin 12 sin co + cos 12 cos co cos i> (13) 

712 = cos co sin i J 

U = sin 12 sin i 1 
m 3 = —cos 12 sin i> (14) 

w 3 = cos i ) 

We require subsequently the following relations between the 
direction cosines: 

Z1WI2 — hmi — n% (15) 

h 2 + U 2 + Zs 2 = 1 (16) 

ffH 2 + m 2 2 + m 3 2 = 1 (17) 

Ixrrii + Z 2 w 2 + hm 3 — (18) 

Any point (£, 1?) on the true orbit projects into the point (x, y) 
on the apparent orbit. We have, consequently, 

x = li% + Z 2 i? 
y = mi£ + m 2 T? 

from which, using Eq. (15), 

.. m& — hy 
? = z. 






n 3 
m\X — l\y 



n 3 
Substituting these values of £ and r\ in Eq. (11), we obtain 

(m 2 x — hy + gens) 2 (mis - Ziy) 2 _ - nQ v 

sw + w u ; 

This is the equation of the apparent orbit and is therefore equiva- 
lent to Eq. (4) (page 75). The coefficients of the same powers 
of the variables in Eqs. (19) and (4) are therefore proportional. 
Denoting the common ratio by /, we have, considering the coeffi- 
cients of x 2 , y 2 , . . . in turn, 

A " tA a 2 & 2 /' ^ 2 \a 2 "*" 6 2 /i 

H = -Jjf 1 ^! + h^l) \ (20) 

n ^A a 2 ^ 6 2 / ( K ' 

q _ fem 2 . p _ feh 

ana ' an* 



84 THE BINARY STARS 

and, from the absolute terms, 

1 = /(e 2 - 1) 
or, writing 

p = a(l - e 2 ) ■ £, (21) 

/ = -| (22) 

This value of / can then be substituted in the expressions of 
Eq. (20). 

We now derive the elements in terms of A, B, . . . F. We 
have, firstly, using Eqs. (21) and (22), 



F 2 - G 2 + A - B = e 



p 2 n s 2 



n* 2^ a / W - Z 2 2 Wl 2 - Zx 2 \ 

(Z 2 2 — W 2 2 ) .1 5 T5 J 

pn 3 2 \ a 2 b 2 ) 

= (*2 2 ~ m**) ( c , + 2)- ( ffl ' 2 ~ W 
p 2 n z 2 \ a/ p 2 n 3 2 

= -J-,0i 8 + ^ 2 - mx 2 - m 2 2 ) 

= ^4?(^3 2 - Z 3 2 ) by Eqs. (16) and (17). 

Hence, using the values of h, ra 3 , n 3 given by Eq. (14), we obtain 
F 2 - G 2 + A - B = ^^ cos 212 (23) 

Again, 

FG — H = e ^ m ^ a_(yni ItmA 

p 2 n s 2 pn 3 2 \ a 2 ap / 

p 2 riz 2 
_ Urrtz 
p 2 n 3 2 

by (18). Hence, using Eq. (14), 

FG - H = -Y^^ sin 212 (24) 

From Eqs. (23) and (24) we obtain 
(F 2 - G 2 + A - B) sin 212 + 2(FG - H) cos 212 = (25) 
which determines 12 in terms of known quantities. 



THE ORBIT OF A VISUAL BINARY STAR 85 

The value of tan 2 i/p 2 can then be found from Eq. (23) or (24). 
Again, it is easily seen that 

F 2 + G 2 - (A + B) = -£- 2 (!x 2 + h 2 + mi 2 + m 2 2 ) 
= -ii(2 - ^ 2 - V) 



1 



,(2 - sin 2 i) 



P' COS" 6 t 

= 2 W< (26) 

P 2 P 2 

Since we already have the value of tan 2 i/p 2 , Eq. (26) enables 
us to calculate p, and therefore i. 

Now, from Eqs. (13) and (20) we find 

G'D COS % 

— m 2 = sin ft sin co — cos ft cos co cos i = (27) 

J5 T 'U COS % 

— l 2 = cos ft sin a) + sin ft cos co cos t = (28) 

Multiply Eqs. (27) and (28) by sin ft and cos ft, respectively, 
and add. Then 

e sin co = p(G sin ft — F cos ft) cos i (29) 

Multiply Eqs. (27) and (28) by cos ft and sin ft, respectively, 
and subtract. Then 

e cos co = -p(G cos ft 4- F sin ft) (30) 

Hence, 

(G sin ft — F cos ft) cos i ,__ 

tan CO = j~ , F . prr — (31) 

(G cos ft + F sin ft) v 

This equation, with Eqs. (29) and (30), determines a without 
ambiguity. 

The eccentricity e may now be obtained from Eq. (29) or 
(30) and the semimajor axis a, from Eq. (21) 

p = a(l - e 2 ) 

To complete the solution analytically, the period P and the time 
of periastron passage T, are to be found from the mean anomalies 
M , computed from the observations by taking the ephemeris 
formulas on page 79 in reverse order. Every M will give an 
equation of the form 



86 THE BINARY STARS 

M = ?~-(t - T), or M = yi + e 

where 

e = -nT. 

From these equations the values of M and T are computed by 
the method of least squares. 

It is more convenient, however, to derive the values of P and T 
from measures on the apparent ellipse made with the aid of a 
planimeter, as follows: The diameter of the apparent ellipse 
drawn through the origin S is obviously the projection of the 
major axis of the true orbit, and the extremity of this diameter 
nearest the origin is therefore the projection of the point of 
periastron. Call it P'. Then, determine c, the constant of 
areal velocity, from planimeter measures of the entire portion 
of the ellipse covered by the observations, or, of that portion of 
this ellipse which seems to be most accurately denned by the 
observations; measure also the areas of two sectors P'Sp and 
P'Sp', p and p' being two observed positions on opposite sides 
of the point P'. Divide these areas by c and apply the quotients, 
with proper signs, to the times corresponding to the positions 
p and p' } and thus derive two values of T, the time of periastron 
passage, which should agree closely. Several sets of points may, 
of course, be used and the mean of all values for T adopted. 
Similarly, the area of the entire ellipse may be measured, and the 
result divided by c to find P, the revolution period. It is not 
necessary to know the unit of area in making these measures, 
since all the areas are simply relative. 

ZWIERS' METHOD* 

Zwiers' method is essentially graphical, and assumes that the 
apparent orbit has been drawn. It may be well to insist again 
that the utmost care must be exercised in drawing this ellipse, 
for unless it gives a good geometrical representation of the obser- 
vations and satisfies the law of areas, it is useless to proceed with 
the orbit computation. 

The apparent ellipse being the projection of the true orbit, its 
diameter drawn through S, the position of the principal star, 
is the projection of the true major axis, and its conjugate, the 

* A. N. 139, 369, 1896. Prof. H. N. Russell independently worked out a 
method based upon the same geometric concept. A. J. 19, 9, 1898. 



THE ORBIT OF A VISUAL BINARY STAR 87 

projection of the true minor axis. Further, if P is that extremity 
of the diameter through S which is nearest S it will be the projec- 
tion of the point of periastron passage in the true orbit. There- 
fore, letting C represent the center of the ellipse, the ratio 
CS/CP will be the eccentricity, e, of that orbit, since ratios are 
not changed by projection. 

Let K = l/\/l — e 2 be the ratio of the major to the minor axis 
in the true orbit; then, if all of the chords in this orbit parallel 
to the minoi axis are increased in the ratio K:l, the ellipse will 
be transformed into Kepler's eccentric circle. Consequently, 
if in the apparent ellipse all ordinates parallel to the conjugate 
diameter, described above, are prolonged in the ratio K:l we 
shall have another conic which may be called the auxiliary 
ellipse. It will evidently be the projection of the eccentric circle. 

The major axis of the auxiliary ellipse will be a diameter of 
the eccentric circle and therefore equal to the major axis of 
the true orbit, and its position will define the line of nodes, 
since the nodal line must be parallel to the only diameter not 
shortened by projection. Designate the semimajor and semi- 
minor axes of the auxiliary ellipse by a and £, respectively; 
then the ratio /3:a is the cosine of the inclination of the orbit 
plane to the plane of projection. Again, the angle a' between 
the major axis of the auxiliary ellipse and the diameter PSCP' 
of the apparent orbit is the projection of the angle w, the angle 
between node and periastron in the true orbit. Therefore 

tan o)' a , /OON 

tan o> = r = — tan co (32) 

cos i j8 

Finally P and T are found by areal measures in the apparent 
ellipse in the manner already described. 

The conjugate diameter required in Zwiers' construction 
may be found most easily by first drawing any chord of the 
ellipse parallel to PSCP', the projected major axis. The 
diameter through the middle point of this chord is the con- 
jugate required. If desired, advantage may also be taken of 
the fact that the conjugate diameter is parallel to the tangents 
to the ellipse at the points P and P', and the rectangular axes 
of the auxiliary ellipse found by trial or by the following con- 
struction : Let 

x 2 . y 2 1 

(a') 2 "•" (&') 2 



88 THE BINARY STARS 

be the equation of the apparent ellipse referred to its conjugate 
diameters. The equation of the auxiliary ellipse referred to 
the same axes will be 

x z y 2 _ 

(a 7 ) 5 + {Kb'f ~ 

The axes are therefore also conjugate diameters of the auxiliary 
ellipse. At the extremity P of the diameter a' (PSCP'), erect 
two perpendiculars, PA and PB, to the tangent to the ellipse 
at this point and make each equal in length to Kb'. Through 
the extremities of the two perpendiculars and the center C of 
the apparent ellipse pass a circle. It will cut the tangent in 
two points, A' and B'. The lines A'C t and B'C will give the 
directions of the two rectangular axes required, the major 
axis lying in the acute, the minor axis in the obtuse angle between 
the diameters a' and Kb'. 

Instead of actually constructing the auxiliary ellipse it will 
generally be easier to derive the elements directly from meas- 
ures of the apparent ellipse with the aid of simple formulas 
based upon the analytical solution of the construction. Thus: 

Let e, a', and b' again represent, respectively, the eccentricity, 
and the projected major and minor axes of the orbit, and let 
x\ and X2 be the position angles of a' and b r . To avoid ambig- 
uity, let Xi be the position angle of the principal star as viewed 
from the center of the apparent ellipse and let X2 be so taken 
that (xi — X2) is an acute angle. Also, compute as before, 
K = 1/V1 — e 2 and b" = Kb'. Then the relations between the 
rectangular axes 2<x and 2/3 of the auxiliary ellipse and the con- 
jugate diameters 2a' and 2b" are given by the equations 

a 2 + £2 = fl /2 _|_ & »2 

a($ = a'b" sin {x x - x 2 ) (33) 

the sine being considered positive. 

The coordinates of any point on the auxiliary ellipse with 
respect to the axes 2a and 2/3 may be written in the form 
a cos 4>', 18 sin 4/. Let a cos (w), £ sin (w) be the coordinates of 
the extremity of the a' diameter; then we shall have 

a' 2 = a 2 cos 2 («) + 2 sin 2 (co) (34) 



and 



la 2 — a' 2 

tan (co) = ±^^—j2 < 35 > 



THE ORBIT OF A VISUAL BINARY STAR 89 

in which the sign of tan (co) is to be the same as that of (xi — x 2 ). 
But co', the projection of w is related to (co) by the equation 

tan co' = - tan (co) (36) 

a 

that is («) = co and ft = (xi — co'). 

The angle co obviously may have either of two values differ- 
ing by 180°; that value is to be taken which will make ft less 
than 180°. 

Zwiers counts all angles in these formulas in the direction of 
increasing position angles. 

The practical procedure may therefore be stated as follows: 
Construct the apparent ellipse and the diameter b' conjugate 
to a'\ measure e, a', V, xi and x 2 ; compute K = l/\/l — e 2 > 
b" = Kb', and find a and @ from 

(a ± j8) 2 = a' 2 + 6" 2 ± 2a'b" sin (xi - X*) 

the sine being taken positive. Then 

a = a 

• 

cos % = - 

a 



, / « 2 - a' 2 
tan co = ±yj a ,i_p > 



the sign of tan co being taken the same as that of (xi — z 2 ), and 
of the two values of co that one which makes S2 less than 180°. 
Next we have 

tan <a' = - tan co £2 = («i — co'), 
a 

and finally deduce the values of P and T from area measurement, 
as in the Glasenapp-Kowalsky method. 

THE THIELE METHOD AND THE THIELE-INNES CONSTANTS 

In 1883,* T. N. Thiele published a method of orbit computa- 
tion depending upon three observed positions and the constant 
of areal velocity, and substituting for the elements a, i, co, and ft, 
the polar coordinates (a, A) and (6, B), referred to the center 
of the apparent ellipse as origin, of two points, P' and R', the 
projections upon the apparent ellipse of the points P and R in 

* A. N. 104, 245, 1883. 



90 THE BINARY STARS 

the true orbit having, respectively, the eccentric anomalies 
0° and 90°. The three observed positions should, of course, be 
normal places carefully selected to represent as long an arc as 
convenient of the well-observed part of the apparent ellipse, 
for the success of the computation depends upon the skill with 
which this selection is made, and the accuracy with which the 
areal constant, defined later, is determined. 

Although the method has a wide range of applicability, it 
did not come in to extensive use until quite recently, when it 
was revived by Innes and van den Bos. Innes, * in 1926, seeking 
a simpler method than those in common use for correcting the 
preliminary elements of an orbit differentially, independently 
developed a method of orbit computation which differs from 
Thiele's only in that he used rectangular instead of polar coordi- 
nates and for Thiele's points R and R' substituted Q, the point 
on the auxiliary circle drawn on the major axis of the true orbit 
that corresponds to R and Q' its projection on the plane of the 
apparent ellipse. If A, B, F, and (?, represent these coordinates, 
the relations between them and Thiele's are : 

Innes Thiele 

A a cos A 

B a sin A 

F b sec <p cos B 

6 b sec <p sin B 

where sin <p — e. 

It will be convenient to treat the two methods together, and 
to follow Innes and van den Bosf rather than Thiele in the 
development of the formulas. 

The data assumed as known are the three normal places 
corresponding to the times ti, U, h and C, the double constant of 
areal velocity. Each observation gives 



= p cos d\ 

= p sin df 



(37) 



These are connected with the usual orbit elements by 

p cos (0 — Q) = r cos (v + w) ) , . 

p sin (0 — Q) == r sin (v + «) cos i) 

* W. H. van den Bos, Orbital Elements of Double Stars, Union Obs., Circ. 
68, 354, 1926; 86, 261, 1932. 
fLoc. cit. 



THE ORBIT OF A VISUAL BINARY STAR 91 

Removing 12 to the right-hand number, and separating v from co, 
we have 



(39) 



x = r cos v (cos w cos 12 — sin co sin 12 cos i) + 

r sin t; (—sin co cos $2 — cos co sin 12 cos i) i 

2/ = r cos w (cos co sin 12 + sin co cos 12 cos i) + 

r sin t> (—sin co sin 12 + cos co cos 12 cos i) t 

It is to be noted that in all the Thiele-Innes formulas the 
inclination i, for retrograde motion, is taken between 90° and 180°. 
Put 



r 
X — - cos v = cos E — sin <p\ 
a ' 

r 
Y = - sin v = cos «> sin 2? 
a 

where sin <p = e 
and 

A = a (cos co cos Q — sin co sin $2 cos i) 

B = a (cos co sin + sin co cos 12 cos i) 

F = a ( — sin co cos 12 — cos co sin U cos i) I 

(r = a ( — sin co sin 12 + cos co cos 12 cos i) , 

Then we have 

x = AX + FF) 
y = 5X + GYf 



(40) 



(41) 



(42) 



Equations (41) show the relations between Innes* constants and 
the elements a, i, co and 12, while Eqs. (42) are his fundamental 
equations. It is also clear that the points (A, B), (F cos v, 
G cos <p), with the center of the apparent ellipse, define a pair of 
conjugate axes which are the projections of the major and 
minor axes of the true orbit. 

The double areal constant is, obviously, twice the area of the 
entire ellipse divided by the period. Since the area of an 
ellipse is t times the product of its semiaxes, and the area of the 
rectangle constructed on the axes equals the area of the parallelo- 
gram constructed on any pair of conjugate axes, we have 

C = ^(AG - BF) cos <p = n(AG - BF) cos <p (43) 

where p = 2t/P — 6.28319/P is the mean annual motion 
expressed in radians. Let A p , q represent the double area of 



92 THE BINARY STARS 

the triangle formed by the position of the primary and the 
positions of the companion at the times t p and t q . Then 

Ap.a = x p y q - x q y p = (AG — BF)(X p Y q — X q Y P ), 

or, introducing values from Eqs. (40) and (43), 

^ = - [sin {E q - E p ) - sin ^(sin E q - sin E p )] (44) 

From Kepler's equation, E — sin (p sin E = M = n(t — T), 
we find 

t q -t p = -[(Eq ~ E P) - Sil1 ^( Sil1 E 1 ~ Sil1 E P)\ 

A 1 

ana, subtracting Eq. (44), we obtain 

t q - t p - ^1 = I [(E q - E p ) - sin (E q - E p )] (45) 

This, in different notation, is Thiele's fundamental equation. 
Let us put E 2 - Ei = u, E 3 - E 2 = v, E z - E x = u + v, 
and write Eq. (45) for (h — ti), (t z - h), and (t z — h). We 
have 

h — h jr = - (u — sin u) 

h-h—^.=-(v- sin v) \ (45a) 

U - h - ^ = - [(u +v) - sin (u + v)]' 

The left hand members of the three equations are known from 
the three normal places and the double areal constant. 

If the observations extend over a full revolution, so that the 
period is approximately known, the approximate value of m 
will be known and we may proceed at once to derive the values 
of e (= sin <p) and E*. If this is not the case, we assume the 
most plausible value for /x, and, with the help of a table for 
(x — sin x), compute u, v and (u + v). By successive approxi- 
mations a value for n should readily be found which will make the 
values of u and v check with the value of (u + v). Then 

D 2x 6.28319 , , 70fl , c 

P = — = , and n = 57.2958m 



THE ORBIT OF A VISUAL BINARY STAR 93 

From Eq. (44) we have, by introducing u, v, (u + v), Ai, 2 , 
A 2 ,3 and A 3 ,i successively, and combining, 

Q 

Ai,2 + A 2 ,3 — Ai,3 = —[sin u + sin v — sin (u + v)] 
/* 

and 

Q 

A 2 ,3 sin u — Ai, 2 sin v = — sin p[sin v(sin E 2 — sin E x ) — 

M 

sin w(sin E% — sin E 2 )\. 

Substituting (2£ 2 — J^i) and (Ez — E 2 ) for u and t> in the right- 
hand member of this equation, removing all parentheses, and 
recombining and substituting, we finally derive 

Q 

A 2>3 sin u — Ai, 2 sin v = — sin <p sin E% [sin u + sin v — sin (u + »)] 

(46) 
Similarly, from 

A 2 ,3 cos u + Ai, 2 cos v — Ai,3 = — sin p[sin Ez — sin Ex 

V- 
— cos w(sin Ez — sm E 3 — cos t>(sin 2? 2 — sin E x )\ 

we derive 

A 2 ,3 cos w + Ai, 2 cos v — Ai,3 = — sin <p cos E 2 

[sin w + sin v — sin (w + v)] (47) 
and from Eqs. (46) and (47) 

. „ A 2 ,3 sin u — Ai, 2 sin v 

sm <p sin E 2 = — 2 — : — 

Ai, 2 + A 2>3 - A 1>3 

«;« „~<, c A 2(3 cos w + Ai, 2 cos v- Ai. s ( *■ ' 

sin ^> COS /i 2 = r ; t r 

Ai, 2 + A 2>3 — A x .3 

These equations give E 2 and e = sin ^; then E\, E 3 follow from 
(E 2 — u) and (E 2 + v). For each time, h, t 2 , t 3 , we next derive 
the mean anomaly from Kepler's equation and thus three values 
of T, the time of periastron passage, which should agree closely. 
The values for X and Y follow from Eq. (40) and, finally, those 
of the four constants, A, F, B, G, from the normal places by the 
use of Eqs. (42). 



94 THE BINARY STARS 

Whatever method of orbit computation is adopted, it is recom- 
mended that the Thiele-Innes constants as well as the elements 
in the usual notation be given. The formulas, to convert from 
the one system to the other, in convenient form for logarithmic 
computation are: 

To derive the Thiele-Innes constants 

A + G = 2a cos (a> + ft) cos 2 - ] 

z 

A — G = 2a cos (w — ft) sin 2 -I 

& * 

B - F = 2a sin (w + ft) cos 2 ^L 

-B - F = 2a sin (o> - ft) sin 2 ^] 
For the inverse process, 

D f 

tan (<o + Q) = A . g 

tan (co - ft) = ~f ~I } (MY 



(49) 



tan 



A -G 

i A-G cos(co + G) -B-F sin(oj + fi) 



2 A+G cos (co-fi) B-F sin (« -ft) 
It may again be pointed out that in these formulas *', in the case of 
retrograde motion, is taken in the second quadrant. 

The practical procedure, then, is first to form the three normal 
places and find the value of C the double areal constant. Thiele 
employs processes of numerical integration for this purpose, 
but it is better, as well as more convenient, to utilize the carefully 
drawn apparent ellipse, or interpolating curves for position angles 
and distances against the times. It is hardly necessary to say 
that the normal places must conform to the law of areas, and, 
that, if they are derived from interpolating curves, care must 
be taken that they fall upon the curve of the apparent ellipse; 

but it may be well to note that C ( = p 2 -^- j has the negative sign 

when the position angles decrease with the time and that the 
units for p and dd/dt are, respectively, seconds of arc, and radians 
per annum. 

* [The tan 2 - formulas are awkward. A better method of solution for a, i, 

to, and CI, is: 

a(l + cosi) = (A + G) sec (co +0) = (B - F) esc (w +0) 
a(l - cosi) = (A - <?)sec(to -Q) = (-B - F) esc (o> -Q), 

from which a and i may be computed easily. — J.T.K.] 



THE ORBIT OF A VISUAL BINARY STAR 



95 



Having the required data, we first find P through the value of n 
derived from Eq. (45a), then e and T, from Eq. (48) and Kepler's 
equation, and finally X and Y and the four constants A, B, F, G, 
from Eqs. (40) and (42). In other words, the three purely 
dynamic elements are derived first, and then values of the four 
constants (representing the geometric elements) to correspond. 

ILLUSTRATIVE EXAMPLES 
In the first edition of this book, I used my computations of the 
orbit of A88 (ADS, 11520) to illustrate the Glasenapp-Kowalsky 



180' 




Fig. 4. — The apparent orbit of A 88. The solidly drawn broken line connects 
the positions used in the computation of the orbit; the dashed and dotted 
lines connect the positions given by later measures. 

method and the method of Zwiers. The period of the system 
is 12.12 years, and the observations now available show that the 
companion has described more than 23^ revolutions about the 
primary since discovery. These later observations, when com- 
pared with an ephemeris based on the orbit, give residuals 
comparable to those obtained for the observations on which the 
orbit was based. I have therefore not revised the elements, 
but retain the original computations which were made in 1912, 



96 



THE BINARY STARS 



Measures and Residuals for A88 



Date 



1900.46 
1901.56 
1902.66 
1903.40 
1904.52 
1905.53 
1906.48 
1907.30 
1908.39 
1909.67 
1910.56 
1911.55 
1912.57 
1914.55 
1914.55 
1915.52 
1916.24 
1916.63 
1916.76 
1917.62 
1917.64 
1918.52 
1918.76 
1919.62 
1920.37 
1920.67 
1921.52 
1921.53 
1922.62 
1923.57 
1923.76 
1924.51 
1924.65 
1925.61 
1928.63 
1931.66 
1932.78 
1933.60 



353-2 
338.3 
318.1 
293.6 
278.4 
224.8 
199.1 
193.5 
178.1 
150.4 
47.0 
18.7 
356.1 
331.2 
330.2 
306.4 
277.2 
243.0 
248.8 
222.5 
228.1 
200.4 
196.9 
188.4 
173.6 
172.6 
143.5 
144.4 
Too close 
10.9 
11.8 
354.9 
344.2 
340.6 
272.2 
187.5 
177.7 
159.3 



Po 



O'.'H 
0.14 
0.12 
0.11 
0.14 
0.12 
0.13 
0.14 
0.15 
0.10 
0.11 
0.15 
0.15 
0.14 
<0.20 
0.15 
0.13 
0.16 
0.14 
0.10 
0.14 
0.14 
0.14 
0.15 
0.16 
0.16 
0.15 
0.12 



0.14 
0.18 
0.15 
0.12 
0.15 
0.14 
0.14 
0.15 
0.11 



Obs. 



Rabe 



VBs 
Lv 



VBs 



VBs 



Btz 
BtF 



Mag 
VBs 



(O-C) 



Ad 



- 1-3 

- 0.8 
+ 0.7 

- 0.2 
+20.8 

- 0.2 

- 7.4 

- 0.8 
+ 0.1 
+ 3.3 

- 0.6 
+ 7.0 
+ 0.9 



+ 
+ 



9.2 
8.2 
+ 10.6 

- 9.8 
-12.1 

- 2.3 

- 3.6 

2. 

7. 

7. 

3 



+ 



- 7.4 

- 2.6 

- 6.9 

- 6.0 

- 2.9 
+ 1.9 

- 2.9 
-11.6 

- 1.5 
+ 12.8 

- 6.9 
+ 2.4 
+ 6.6 



-0'-'004 
-0002 
+0.002 
-0.000 
+0.040 
-0.000 
-0.020 
-0.002 
+0.000 
+0.006 
-0.001 
+0.017 
+0.003 
+0.022 
+0.020 
+0.020 
-0.017 
-0.023 
-0.004 
-0.008 
+0.008 
-0.021 
-0.020 
-0.009 
-0.020 
-0.007 
-0.011 
-0.009 

-0.007 
+0.005 
-0.009 
-0.034 
-0.004 
+0.025 
-0.020 
+0.006 
+0.012 



A P 



-0'.'03 
-0.02 
-0.01 

0.00 
+0.03 
+0.01 
-0.03 
-0.03 

0.00 
+0.01 
+0.03 
+0.01 
-0.02 

0.00 

+0.04 
+0.03 
+0.05 
+0.03 
+0.03 
+0.01 
-0.02 
-0.02 
-0.02 
0.00 
+0.01 
+0.06 
+0.03 

0.00 
+0.03 
-0.02 
-0.05 
-0.01 
+0.03 
-0.03 

0.00 
+0.01 



The average residual in angle expressed in radians, A0(p/57.3), is ±0''012; and in dis- 
tance, Ap, ±0'/022. 



THE ORBIT OF A VISUAL BINARY STAR 97 

adding, however, the later measures and the corresponding 
residuals. 

All of the measures to date are given in the table on page 96. 
The dates, observed position angles, and observed distances are 
recorded in the first three columns. The fourth column shows 
the number of measures (on different nights) on which each posi- 
tion rests. Almost all of the measures were made by me, with 
the 36-in. refractor; for the others, the name or an abbreviation 
of the name of the observer is entered in the fifth column. 
(VBs = Van Biesbroeck, Lv = Leavenworth, Btz = Bernewitz, 
Mag = Maggini.) The last three columns give the residuals, 
observed minus computed, for the position angles and distances, 
the former entered both in degrees and reduced to stricter 
comparison with the distance residuals by multiplying by the 
factor p/57.3. 

All of the measures to 1912 inclusive were plotted, using a 
scale of 3 in. to 0"1, and, after repeated trials, the ellipse shown 
in the diagram on page 95 was drawn. It represents the observa- 
tion points fairly and satisfies the law of areas closely. Applying 
the Glasenapp-Kowalsky method, we first measure the intercepts 
of the ellipse with the axes of coordinates, and the coordinates of 
two selected points for the value of H, counting the end of the 
a>axis at 0°, and of the t/-axis at 90°, positive. The measures 
are (in inches on the original drawing) : 

xi = +4.98, ?/i = +1.77, x a = -2.55, y a = -2.86 
Xi = -4.73, y* = -3.12. x b = +3.17, y b = -2.49 

Therefore we have 

xix 2 = -23.5554, y x y 2 = -5.5224, x a y a = +7.2930, 

xi + x 2 = +0.25, yi + y 2 = -1.35, x b y h = -7.8933, 

xj = 6.5025, y a 2 = 8.1796 

Xb 2 = 10.0489, y b 2 = 6.2001 

from which to compute the five constants of the equation of 
the ellipse. We find 

A = — = -0.04245 

#1#2 

B = — = -0.18108 

F = - y \ + y * = -0.12223 
22/i2/2 



98 THE BINARY STARS 

G = _ *i+ x * = +0.00531 
2zi£ 2 

From these values and the coordinates x a , y a , we obtain 

H = - Ax * + B * + ™ x + ^ + 1 = +0.00584, 

2xy 

and, similarly, from the coordinates Xb, yb, 

H = +0.00590, 

and adopt the mean, +0.00587. 

Combining these constants, we have, 
FG = -0.00065; F 2 = +0.01494; G 2 = +0.00003; 
-2(FG - H) = +0.01304; F 2 - G 2 + A - B = +0.15354; 
F 2 + G 2 - (A + B) = +0.23850. 

The solution of Eqs. (24), (23), (26), (29), (30), and (21) 
then proceeds as follows : 

log i ^l sin 2fl 8.11528 
p 2 

log i ^J cos 212 9.18622 

log tan 2S1 8.92906 

2 fl 4?85 

fi 2?42 

log cos 212 9.99844 

log ***£* 9.18778 



tan 2 * 



+0.15409 



9 ton 2 i" 

From Eq. (26) - 2 + --^ +0.23850 

2 



J. 

»2 



+0.08441 
+0.04220 

log - a 8.62536 

log p 2 1.37464 

log v 0.68732 

log tan 2 1 0.56242 

log tan i 0.28121 

180° - i = 62°22'30" = 62 °A 



THE ORBIT OF A VISUAL BINARY STAR 



99 



log 



logF 
sin 12 
cos 12 

(1) log F cos 12 

(2) log F sin $2 

(1) 
(2) 

(3) - (1) 

log [(3) - (1)] 

cos i 

logp 

log e sin u 

log tan co 

CO 

sin a 

log e 

e 

e 2 

1 - e 2 

log (1 - e 2 ) 

logp 

V 



log 
9.08718 n 
8.62557 
9.99962 
9.08680 n 
7.71275 n 
-0.12212 
-0.005161 
+0.12234 
9.08757 
9.66586 
0.68732 
9.44075 
2.59507* 
90?1 
0.00000 
9.44075 
0.276 
0.07618 
0.92382 
9.96559 
0.68732 



(1 - e>) 



= log a 0.72173 

a 5.269 in. 
= 0^176 



i 117?6 
cos i 9.66586 

logG 
sin J2 
cos $2 

(3) log G sin J2 

(4) log G cos 12 

(3) 
(4) 

(2) + (4) 

log - [(2) + (4)] 

V 



7.72509 

8.62557 

9.99962 

6.35066 

7.72470 

+0.00022 

+0.005305 

+0.000144 

6.15836„ 

0.68732 



log e cos co 6.84568n 



From the diagram. (Fig. 4) it is obvious that the companion 
passed its periastron point between the dates of observation 
1909.67 and 1910.56; but the measures made in 1908 and 1912 
were regarded as more reliable than these and were accordingly- 
used to determine the time of periastron passage. The constant 
of areal velocity (in units of the planimeter scale) had been found 
to be 0.205. Drawing radii to the points P and 1908.39 and 
1912.57, the areas of the two resulting sectors were, in terms 
of the same unit, respectively, 0.34440 and 0.50225. Hence 
the time intervals between these two dates and the date of 
periastron passage were, respectively, +1.68 years, and —2.45 
years, giving for T, the two values, 1910.07 and 1910.12. The 
mean, 1910.1, was adopted. The planimeter measures gave as 



100 THE BINARY STARS 

the area of the entire ellipse, 2.4848, and the period, 12.12 
years. 

To solve the orbit by Zwiers' method, we begin by finding 
the axis b' conjugate to a' (PSCP'). Draw the chord cc par- 
allel to P'CSP and then draw the diameter through its middle 
point. This will be the required conjugate. 

We now measure CS = 0.67, CP = 2.45, a' = 2.445, b' = 5.050; 
and the angles Xi = 92°6 and x 2 = 3°6. 

The ratio CS:CP gives at once the value of the eccentricity, 
e = 0.273, and from this we compute the value of K = \/\/l — e 2 
(in logarithms) 0.01682. Thence we find b" = Kb' = 5.2494. 

The computation then proceeds as follows: 



log a' 


0.38828 


log b" 


0.72011 


log 2 


0.30103 


log sin (xi — X2) 


9.99993 


log 2a'b" sin (xi — x 2 ) 


1.40935 


2a'b" sin (x% — x 2 ) 


25.6653 


(a') 2 + (b"Y 


33.5342 


(a + £> 2 


59.1995 


(a " PY 


7.8689 


(a+/8) 


7.6942 


(«-/8) 


2.8052 


2a 


10.4994 


20 


4.8890 


a 


5.2497 


P 


2.4445 


log/3 


0.38819 


log a 


0.72014 


log cos i 


9.66805 


.'. i = 


62°25 


a = a = 


5.25 in. 




(tf'175 


(a')« 


5.9780 


(6") 2 


27.5562 


[(a') 2 + (6") 2 ] 


33.5342 


a 2 


27.5600 


2 


5.9756 


a 2 - (a') 2 


21.5820 



THE ORBIT OF A VISUAL BINARY STAR 101 



(«') 2 - p 


0.0024 


log [a 2 - (a') 2 ] 


1.33409 


log [(a') 2 - J8 2 ] 


7.38021 


log tan 2 w 


3.95388 


log tan w 


1.97694 


.'. u = 


= 89?4 


log cos i 


9.66805 


log tan «' 


1.64499 


«' 


88^7 



/. = (xi - «') = 3?9 
Assembling the elements we have the following : 

Glasenapp's Method Zwiers' Method 



P= 12.12 years 


12 . 12 years 


T = 1910.10 


1910.10 


e = 0.276 


0.273 


a = 0"176 


, . , 175 


w = 269^9 


270^6 


i = 117?6 


117?75 


Q = 2.4 


3.9 



Angles decreasing with the time. 

In the formulas, all angles are counted in the direction of 
increasing position angles, whereas in the notation given on 
page 78, « is counted from node to periastron in the direction 
of motion of the companion. Therefore, when as in this system 
the. observed position angles decrease with advancing time, the 
value for « derived from the formulae must be subtracted from 
360°. In applying the formulas for computing the ephemeris 
of such a system, the anomalies are counted positive after 
periastron passage and negative before, just as in the case of 
direct motion (angles increasing with the time) ; cos i is counted 
as positive, and the angles (0 — fi) are taken in the quadrant 
360° — (v + co). I have found this to be the simplest and most 
satisfactory method of procedure in every case where the angles 
decrease with the time. In orbits with direct motion the value 
of « is used as given directly by the formulas and the angles 
(6 — fl) are taken in the same quadrant as the angles (« + «). 

The orbit of 24 Aquarii* has been selected as an illustration of 
Thiele's method, as it is a beautiful example of the power of that 

*24 Aqr = B 1212 = BDS 11125 = ADS 15176. 



102 



THE BINARY STARS 



method in dealing with a rather recalcitrant case. The orbit 
was computed by W. S. Finsen and is published in Union Observa- 
tory Circular 81, 112, 1929. In the list of measures on page 
103 the columns give respectively the date, the position angle, the 
distance, the number of nights, the aperture, the observer, and 
the residuals (observed minus computed) in angle and in distance. 
An asterisk attached to the date denotes measures overlooked 
in Finsen's compilation or not available at the time. 



1930 




Scale 
Fig. 5. — The apparent orbit of 24 Aquarii. 

The angles and distances were plotted against the time and 
interpolation curves drawn. In the years 1922-1924 the meas- 
ures are discordant and uncertain; they were therefore dis- 
regarded, the areal constant being based on the arc 1891-1918 
alone and the three normal places required being taken suffi- 
ciently far away from the unreliable part of the interpolation 
curves. 

The angles and distances were read for every second year as 
shown in the table on page 104. 

The numbers in the column headed did 2 Ap, giving the products 
of two successive distances and the sector-angle between them, 
may be considered sufficiently close approximations to the double 
areas of the sectors (provided that the distance does not vary too 
greatly within the sector) and should therefore be constant. 



THE ORBIT OF A VISUAL BINARY STAR 
Measures and Residuals for 24 Aquarii 



103 











Tel. 


Obs. 


O - 


- C 


Date 


00 


PO 


n 


















A9 


Ap 


1890.75 


254?5 


0745 


3 


36 





- 2.0 


-0708 


1.75 


261.0 


0.55 


4 


36 





+ 4.0 


+0.01 


2.40 


256.2 


0.38 


2 


19 


Sp 


- 3.0 


-0.16 


3.68 


260.5 


0.55 


3 


16 


HCW 


- 0.6 


0.00 


3.88 


262.8 


0.59 


1 


36 


Bar 


+ 1.4 


+0.04 


4.82 


264.7 


0.52 


7 


36 


Bar 


+ 1.9 


-0.03 


4.86 


261.5 


0.45 


3 


19 


Sp 


- 1.3 


-0.10 


7.81 


263.5 


0.65 


3 


36 


A 


- 3.6 


+0.09 


7.89 


267.4 


0.73 


1 


26 


SBn 


+ 0.2 


+0.17 


8.78 


269.0 


0.49 


3 


36, 12 


A 


+ 1.7 


-0.07 


8.84 


269.0 


0.54 


1 


40 





+ 0.4 


-0.02 


9.98 


282.6 


0.5e 


1 


26 


SBn 






1900.67 


205.1 


0.5e 


1 


6 


Sola 






0.74 


271.9 


0.55 


2 


14 


Dob 


+ 0.7 


-0.01 


1.54 


269.4 


0.49 


10 


28 


GrO 


- 3.0 


-0.07 


1.79 


274.0 


0.55 


2 


36 


A 


+ 1.2 


-0.01 


2.00 


273.0 


0.57 


11 


18 


Doo 


- 0.1 


+0.01 


3.86 


273.8 


0.54 


2 


15 


VBs 


- 1.9 


-0.01 


4.54* 


273.0 


0.48 


5 


15M 


Com 


- 3.7 


-0.07 


4.67 


278.6 


0.49 


1 


36 


A 


+ 1.7 


-0.06 


8.35 


282.6 


0.52 


3 


15K 


Com 


- 0.1 


0.00 


8.47 


285.5 


0.49 


4 


15 


VBs 


+ 2.7 


-0.03 


8.72 


279.6 


0.68 


2 


26 


Ol 


- 3.6 


+0.16 


8.72 


284.8 


0.56 


2 


26 


RW 


+ 1.6 


+0.04 


8.73 


286.4 


0.72 


2 


26 


Neff 


+ 3.2 


+0.20 


9.85 


281.1 


0.58 


3 


18 


Doo 


- 4.1 


+0.07 


1910.40 


282.2 


0.51 


3 


15H 


Com 


- 3.9 


+0.01 


0.72 


278.2 


0.43 


5 


28 


GrO 


- 8.5 


-0.07 


1.20 


291.9 


0.52 


3 


6 


Dob 


+ 4.3 


+0.02 


1.68 


286.8 


0.50 


3 


15 


VBs 


- 1.6 


+0.01 


4.00 


292.5 


0.47 


8 


28 


GrO 


- 0.5 


+0.01 


4.63 


291.3 


0.47 


2 


36 


A 


- 2.7 


+0.02 


4.65 


294.6 


0.48 


2 


15^ 


Com 


+ 0.6 


+0.03 


4.66* 


293.5 


0.51 


1 


40 


Lv 


- 0.5 


+0.06 


4.79* 


298.1 


0.45 


3 


10M 


Lv 


+ 3.6 


0.00 


4.94 


311.1 


0.41 


5 


8 


Rabe 


+ 16.0 


-0.04 


6.42 


296.5 


0.53 


3 


26 


Ol 


- 2.0 


+0.11 


6.62 


293.1 


0.46 


3 


12,40 


Lv 


- 5.9 


+0.04 


6.65 


301.9 


0.41 


3 


15H 


Com 


+ 2.9 


-0.01 


7.69 


305.9 


0.40 


3 


15J^ 


Com 


+ 3.9 


0.00 


7.74 


294.7 


0.42 


1 


40 


VBs 


- 7.5 


+0.01 


1921.66 


321.1 


0.22 


3 


36 


A 


+ 3.3 


-0.07 


2.81 


40 ? 


<0.1 


1 


36 


A 






3.62 


341.8 


0.20 


2 


40 


VBs 


+ 11.3 


-0.01 


3.85 


6.0 


0.57 


4,3 


30 


Bail 






3.88 


333.5 


0.15 


3 


13 


Mag 


+ 0.3 


-0.05 


4.55 


55 


0.12 


1 


36 


A 




-0.04 


4.71 


6.9 


0.22 


1 


30 


Plq 




+0.07 


4.82 


350.0 


0.16 


1 


40 


VBs 


+ 2.1 


+0.02 


1926.64 


190.7 


0.20 


1 


261-6 


B 


-10.0 


+0.05 


6.69 


204.2 


0.19 


1 


36 


A 


+ 2.9 


+0.03 


7.74 


2H.0 


0.21 


3 


26M 


B 


- 3.6 


-0.02 


7.74 


218.7 


0.23 


1 


26H 


<P 


+ 4.1 


0.00 


8.73* 


224.6 


0.26 


1 


36 


A 


+ 2.9 


-0.01 


8.75 


221.2 


0.27 


4 


26M 


B 


- 0.6 


0.00 


8.75 


222.6 


0.26 


4 


26H 


<p 


+ 0.8 


-0.01 


9.46* 


228.8 


0.28 


4 


23>£ 


V 


+ 3.1 


-0.02 


9.63* 


230.2 


0.27 


1 


36 


A 


+ 3.8 


-0.04 


9.86* 


227.8 


0.26 


3 


26W 


B 


+ 0.3 


-0.05 


1930.48* 


234.3 


0.29 


3 


23>| 


V 


+ 4.3 


-0.04 


1.66* 


236.0 


0.37 


2 


36 


A 


+ 1.6 


0.00 


2.79* 


236.9 


0.35 


4 


26K 


B 


- 0.9 


-0.04 


2.79* 


238.3 


0.30 


1 


2GM 


<p 


+ 0.5 


-0.09 



The actual figures show the need of further adjustment. As 
this can be made both for Ap and for d, it is advisable to plot the 
polar coordinates as a check that the adjusted positions show a 



104 



THE BINARY STARS 



smooth elliptic arc. In general it is found that the major part 
of the adjustment has to be made in the distances. 

The columns on the right of the double rule show the result of 
the adjustment, which may be regarded as satisfactory. 



t 


V 


Ap 


d 


didzAp 


V 


Ap 


d 


d\.d%Ap 


1890 


255?5 


+3?1 


0'.'47 


+0.71 


255?4 


+3?2 


0"53 


+0.92 


2 


258.6 


2.9 


0.49 


0.74 


258.6 


3.0 


0.54 


0.89 


4 


261.5 


2.5 


0.52 


0.70 


261.6 


2.9 


0.55 


0.89 


6 


264.0 


2.0 


0.54 


0.59 


264.5 


2.8 


0.56 


0.89 


8 


266.0 


3.0 


0.55 


0.92 


267.3 


2.7 


0.57 


0.88 


1900 


269.0 


3.0 


0.56 


0.94 


270.0 


2.8 


0.57 


0.91 


2 


272.0 


3.6 


0.56 


1.15 


272.8 


2.9 


0.57 


0.92 


4 


275.6 


2.6 


0.57 


0.85 


275.7 


3.0 


0.56 


0.92 


6 


278.2 


3.5 


0.57 


1.14 


278.7 


3.2 


0.55 


0.93 


8 


281.7 


3.6 


0.57 


1.11 


281.9 


3.4 


0.53 


0.92 


10 


285.3 


3.0 


0.54 


0.84 


285.3 


3.6 


0.51 


0.90 


12 


288.3 


3.9 


0.52 


0.99 


288.9 


4.0 


0.49 


0.90 


14 


292.2 


4.8 


0.49 


1.06 


292.9 


4.6 


0.46 


0.91 


16 


297.0 


6.1 


0.45 


1.10 


297.5 


5.5 


0.43 


0.92 


18 


303.1 




0.40 
Mean 




303.0 




0.39 
Mean 






+0.917 


+0.907 



The mean value of did 2 Ap, +0.907, has now to be divided by 
the interval in time, here 2, and by 57.3 to reduce Ap to radians. 
We obtain for the double areal constant: c = +0.007914. 

As the result of a similar adjustment Finsen found c = +0.00781 
which he supplemented by the three normal places 



THE ORBIT OF A VISUAL BINARY STAR 105 



1892.00 


258? 6 


1910.00 


285? 3 


1928.00 


21 T- 



0'54 
0"51 
0"24 



The above adjustment, which is equivalent to, though experts 
in the method regard it as considerably simpler than, the con- 
struction of the apparent orbit in the graphical methods, has been 
given rather fully, for, as noted in describing the method, it is 
the crux of the problem. It is at this stage that the computer's 
judgment, experience, and knowledge of the reliability of the 
heterogeneous observational material come into play; once 
the three normal places and the double areal constant have been 
found, the derivation of the orbital elements is simple and 
straightforward. 

It sometimes happens that the period is known a priori 
(from the recurrence of position angle) while the character of the 
apparent orbit, or large gaps in the observed arc, cause difficulties 
in the determination of c. It is therefore worth remembering 
that in Thiele's fundamental formula 

t q - t P - — = - [E q -E p -sw (E q - E p )] 

we may regard either /torcas unknown. 

Adopting Finsen's normal places and c so as to obtain his 
elements, the computation proceeds as follows: 

U = 1892.00 xi = -0.107 y x = -0.529 A w = +0.1240 
t 2 = 1910.00 x 2 = +0.135 y 2 = -0.492 A 2t3 = -0.1139 
t 3 = 1928.00 x 3 = -0.192 y z = -0.144 Ai., = -0.0862 

^ = +15.88 u - sin u = 2.12** 

c 

^ = -14.58 t; - sin v = 32.58/t 

c 

^ = -11.04 (u + v) - sin(w + v) = 47.04m 
c 

In the approximations on page 106 for /* the table for x — 
sin x given in Union Observatory Circular No. 86 is used; it is 
helpful to plot the differences found in the last row against the 
values of n in the first row, as after three approximations have 
been made the resulting differences frequently define the curve 
so well, that the fourth approximation becomes final. 



106 



THE BINARY STARS 



As we expect the period to be of the order of 50 years (/x = 0. 126) , 
we try n = 0.12 and /* = 0.13: 



u — sin u 

v — sin v 

(i* + v ) — sin (u + v) . 
u 



v 

Sum 

(u +»)... 
Difference. 



0.12 



0.2544 

3.9096 

5.6448 

67.51 

202.27 

269.78 

266.25 

+3.53 



0.13 



0.2756 
4.2354 
6.1152 
69.43 
212.17 
281.60 
301.54 
-19.94 



0.122 0.1224 



0. 

3. 

5. 
67. 
204. 
272. 
271. 
+0. 



2586 

9748 

7389 

90 

22 

12 

53 

59 



0.2595 
3.9878 
5.7577 
67.98 

204.61 

272.59 

272.66 

-0.07 



and find 

M = 0.1224 u = 67?98 v = 204?61 
Finsen, working to another decimal place, obtained : 

H = 0.12241 u = 68?022 v = 204?642 Final difference 

-0?002 
and hence for period and mean motion: 

P = 51.33 years n = 7?0134 

Further 

e sin E 2 = -0.5598 e cos E 2 = -0.7178 

e = 0.9102 E 2 = 217?95 

tf i = E 2 - u = 149?93 E 3 = E 2 + v = 62?59 

{Mi 123?80 
M 2 250?02 
M 3 16?29 

all giving T = 1925.68. 

X x = -1.7754 Y t = +0.2078 
X 2 = -1.6985 F 2 = -0.2550 
X 3 = -0.4492 Y 3 = +0.3679 

From the first and third normal places 

A = -O'.'OOll, B = +0'. , 2939, F = -0 , . , 5233, G = -0'.'0317. 

These values give 

x 2 = +0'.'135 y 2 = -O'.^Ol 

for the second normal place, a satisfactory check. 



THE ORBIT OF A VISUAL BINARY STAR 107 

Though the conflicting results obtained in the years 1922-1924 
were entirely disregarded in deriving the orbit, partly because 
it was not even possible to assign the quadrants with certainty, 
the residuals show that the measures by Aitken in 1921, Maggini 
in 1923, and van Biesbroeck in 1923 and 1924 are satisfactorily 
represented. This success of the orbit in bridging the large gap 
from 300° to 200°, a sector of 260° in all, inspires confidence in 
the elements found. 

A = -0.0011 B = +0.2939 

G = -0.0317 F = -0.5233 

A + G = -0.0328 B -F = +0.8172 

A - G = +0.0306 -B -F = +0.2294 

cos (co + 12) negative, sine positive 
cos (co — 12) positive, sine positive 

I q gi 79 
tan (co + 12) = _ ' 2g = -24.9 second quadrant 

tan <- - a > - £nfs - + 7 - 49 fir8t " 

co + Q = 92?30 
to - 12 = 82.40 

co = 87?35 J2 = 4?95 2co = 174.70 

212 = 9.90 

If 12 were found to be in the third or fourth quadrant, 180° 
should be added to (or subtracted from) both co and 12. 

cos (co + 12) = -0.0401 sin (co + 12) = +0.9992 
cos (co - 12) = +0.1323 sin (co - 12) = +0.9912 

x 9 i +0.2294 +0.9992 OQQn i , nW9n 

tan 2 = +63172 ' +099T2 = + - 2830 tan 2 = ±0 ' tSa2 ° 

| = ±28?01 % = ±56?02 cos | = +0.8828 

a - B ~ F - ° 8172 = n-525 

~ 7T^ 7~Z ~i ~ 2 X 0.9992 X 0.7793 u * 
2 sm(co + 12) cos 2 K 

Assembling the elements in the usual notation and the Thiele- 
Innes constants, we have: 



108 



THE BINARY STARS 



P = 

T = 

a = 

e = 

« = 

i = 

S2 = 

A = 

B = 

F = 

G = 



51.33 years 

1925.68 

0'/525 

0.9102 

87?35 
±56?02 
4?95 
-0.0011 
+0.2939 
-0.5233 
-0.0317 



CONSTRUCTION OF THE APPARENT ELLIPSE FROM THE 
ELEMENTS 

It is sometimes desirable to be able to construct the apparent 
ellipse from the elements of the true orbit. This construction 
is easily and quickly effected in the following manner: 




Fig. 6. — The true and apparent orbits of a double star. (After See.) 

Take the point (Fig. 6), at the intersection of two rectangular 
axes, OX and OY, as the common center of the true and pro- 
jected orbits. Draw the line OQ making an angle equal to 12 
with the line OX, counting from 0°. Lay off the angle <a from 
the line Ofi, starting from the extremity 12 between 0° and 180° 



THE ORBIT OF A VISUAL BINARY STAR 109 

and proceeding in the direction of the companion's motion (clock- 
wise, that is, if the position angles decrease with the time, 
counterclockwise, if they increase with the time). This will 
give the direction of the line of apsides, AOP, in the true orbit. 
Upon this line lay off OS, equal to ae, the product of the eccen- 
tricity and the semiaxis of the orbit, using any convenient 
scale, and OP and OA, each equal to a. The point S lies between 
and P, and P is to be taken in the quadrant given by applying 
(a to $2 as described above. Having thus the major axis and the 
eccentricity, the true ellipse is constructed in the usual manner. 

Now divide the diameter yOO of this ellipse into any con- 
venient number of parts, making the points of division symmet- 
rical with respect to 0, and draw chords 6/36', etc., perpendicular 
to the line of nodes. Measure the segments /36, 06', etc., and 
multiply the results by cos i. The products will evidently be 
the lengths of the corresponding segments /9&i, fib 2 , etc., in the 
projected ellipse, and the curve drawn through the points &i, 
C\, di, . . . will be the desired apparent orbit. 

To find the position of the principal star in the apparent 
ellipse draw through S a line perpendicular to the line of nodes, 
and on it lay off the length ae sin a> cos i from the line of nodes. 
This will give S', the point required. Lines through S' parallel to 
OX and OY will be the rectangular axes to which position angles in 
the apparent orbit are referred, and the position angle of the com- 
panion at any particular epoch may be obtained by laving off the 
observed position angle. The line OS' extended to meet the ellipse 
demies P', the projection of the point of periastron passage. 



DIFFERENTIAL CORRECTIONS 

If sufficient care is exercised in the construction of the apparent 
ellipse, methods like those described will, as a rule, give a pre- 
liminary orbit that will satisfy the observed positions within 
reasonable limits and that will approximate the real orbit 
closely enough to serve as the basis for a least squares solution. 
It should be emphasized that a satisfactory representation 
of the observed positions does not necessarily imply a correct 
orbit when the arc covered by the observations is comparatively 
small. The percentage of error inherent in double star measures 
is so great that, if the observed arc is less than 180°, it will 
generally be possible to draw several very different ellipses 



110 THE BINARY STARS 

each of which will satisfy the data of observation about equally- 
well. In general, it is not worth while to compute the orbit of a 
double star until the observed arc not only exceeds 180°, but also 
defines both ends of the apparent ellipse. 

It may not be amiss to add a few words on the practice, all 
too common, of giving a fictitious appearance of accuracy to the 
values of the prehminary elements by the use of unwarranted 
decimals. To give the angular elements, the eccentricity, or 
the semimajor axis to three or four decimals when even the first 
decimal is uncertain, or the period and time of periastron passage 
to the second or third decimal when the latter may be in doubt 
by years and the former by decades, or in extreme cases, by 
centuries, adds nothing to the real accuracy of the results and 
does not inspire confidence in them. 

Many computers are content with a preliminary orbit; but 
it is advisable to correct these elements by the method of least 
squares whenever the data are sufficient for an investigation 
of the systematic errors of observation. 

The position angle is a function of the six elements 

QAf)° 

ft, i, a?, e ( = sin <£), T and n = — =- 

and the required differential coefficients for the equations of 
condition can be computed with all necessary accuracy from 
the approximate formula 

AAft + BAo> + CM + DA<p -f FAM + GA M + (C - 0) = (51) 

where Aft, Aoj, etc., are the desired corrections to the elements, 
and (C — 0) is the difference between the position angle com- 
puted from the preliminary elements and the observed angle 
and A, B f etc., are the partial differential coefficients. The 
eccentric angle <p, defined by sin <p = e, and the mean anomaly 
M [= fi(t — T)] are substituted for the eccentricity (e) and the 
time of periastron passage (T). 

The corresponding equation for the distances may be written 

hAa + bAco + cAi + dA<p + fAM + gAfi + (C - 0) = (52) 

the difference between the computed and observed distance 
forming the absolute term. 

The values of the partial differential coefficients are derived 
from the equations for 6 and p (pages 79, 80). It has been the 



THE ORBIT OF A VISUAL BINARY STAR 111 

common practice to base the corrections to all the elements 
except the semimajor axis, solely upon the residuals for position 
angle, but it is sounder, theoretically at least, and in many cases 
practically as well, to utilize the residuals both in angle and in 
distance for the corrections to the five elements w, i, <p, M and n 
by combining the two sets of equations of condition after elimi- 
nating A £2 from the first set and Aa from the second. If this 
combination is made, the term (C — 0) in the equations for 
position angle must be expressed in circular measure, which is 
accomplished by multiplying the angle residuals by the factor 
p/57.3. 

Comstock has put the expressions for the differential coeffi- 
cients into the form given below. The residuals in angle are 
assumed to be expressed in circular measure, and the expressions 
are so formulated that the solution of the equations of condition 
by the method of least squares will give the correction to the 
semimajor axis in seconds of arc, and those to the other elements 
in degrees. 

Introduce the auxiliary quantities m, k and a defined by 
m = p/57.3 = (8.2419)p, 

k = (2 + sin cp cos v) sin E, and 

a = — m tan * sin (0 — Q.) cos (0 — £2). 

The differential coefficients for the terms given in the central 
column are: 

Angle Terms Distance 

A = +m A£2 

B = +wf - | cos i Aa> b = + <r sin i 

W 
C = +<r At c = +<r tan (0 — Q) 

D = +.B(- \k A<p d = -\-b(-\k — ml- J cos <p cos v 

F = —El-] cos^j AM /= — b( - J 2 cos^ — ml - J sin^>sin.E 
G = - F (jt - T ) An g = -f(t - T) 

Aa h = -\ — 
a 

If the Thiele-Innes constants A, B, F, G, have been computed, 
the following method will be found convenient, particularly if 



112 THE BINARY STARS 

tables for X and Y are available.* The equations of condition 
take the form 

Ax = XAA + YAF + P x Ae + Q x nAT + R x An\ 

Ay = XAB + YAG + P y Ae + Q^nAT 7 + ^Aw/ { 6) 

where P* = +A^ + F^ 
de de 

p > = +*%+<€ 

V * " *dM ^dM 
B x = -{t - !T)Qs 
i2 v = -(«- TOQv 

From the formulas already given, 

M = E — e sin E 
X = cos E — e 

F = Vl - e 2 sin M = - sin v J 

e = sin <p, \/l — e 2 = cos ^ 

we have, by differentiation, 

dX 



— = -OOlfl -\ — 

de I cos 2 ^>(cos 2 <p — X sin ^>) 

(54) 



*L = looi ZF 

de ' cos 2 ^>(cos 2 ^ — X sin ^>) 

|^ = -0.017453 , j-^ — ^ c 

aAf cos ??(cos 2 <p — X sin ^) 

•g = +0.017453 COS / (X + 8in y j 
aiM cos 2 <p — X sin ^>) 

the units being 0.01 for Ae and 1° for AM. 

The question of algebraic sign is always an important one in 
computing differential coefficients. For the present formulas 
van den Bos gives the following rules : 

dX/de is always negative, 

* Such tables have been issued, in an Appendix to Un. Obs. Circ. 71. The 
method described is given in Circs. 68 and 86. 



THE ORBIT OF A VISUAL BINARY STAR 113 

dY/de is positive when X and Y have the same sign, negative 
when they have opposite signs, 

dX/dM has the sign contrary to that of Y, 

dY/dM is negative only when X is negative and numerically 
greater than e. 

These rules follow from the fact that cos <p and the expression 
(cos 2 <p — X sin <p) — which is simply a transformation of the 
expression 1 — e cos E derived by differentiating Kepler's 
equation — are always positive, and, for the last two formulas, 
from simple geometrical considerations. The sign of X is given 
directly in the tables referred to; that of Y is always the same as 
that of M , the mean anomaly. 

The special advantage of the method arises from the fact that 
the values of dX/de, etc., can be read off directly from the tables 
of X and Y, for the value of the differential coefficient differs 
from the mean of the preceding and following first difference, 
in the tables (taken horizontally, for de and vertically for dM), 
by only one-sixth of the third difference, a negligible quantity. 
If these tables are not available, the differential expressions 
dX/de etc., may be computed from the formulas given above, 
special attention being given to the units for Ae and AM and to 
the algebraic signs. 

SPECIAL CASES 

All methods based upon the construction of the apparent 
ellipse fail when the inclination of the orbit plane is (within 
the limit of error of observation) 90°; for the apparent ellipse is 
then reduced to a straight line and the observed motion is entirely 
in the distance, the position angle remaining constant except for 
the change of- 180° after each occultation or apparent merging 
of the two components into a single image. Such a Hmiting 
case is actually presented by 42 Comae Berenices (21728). 
Many other pairs are known in which the orbit inclination is 
only slightly smaller. 

In the hmiting case, the elements, except 12 which is obviously 
given by the observed position angle, and i (90°), must be 
derived entirely from the observed distances; in other cases 
special methods may be devised which will vary with the pecul- 
iarities of the observed motion but which will depend in large 
part upon the observed distance. 



114 



THE BINARY STARS 



The first orbit for 42 Comae Berenices was published by Otto 
Struve in 1875,* but his paper contains no statement as to the 
methods used in obtaining his preliminary set of elements, and, 
in 1918, 1 was unable to find in print any solution of the problem. 
Simple graphical methods for finding the elements P, T, and e 
from the curve of observed distances at once suggested them- 
selves; but methods for deriving the values of a and w were not 
immediately apparent. In the first edition of this book I 
outlined a method for obtaining them which Prof. F. R. Moulton 
had kindly sent me in manuscript. More recently other methods 
have been published by Prof. Kurt Laves, f Dr. F. C. Henroteaut 




Fig. 7. 



-Apparent and true orbits and interpolating curve of observed distances 
for a binary system in which the inclination is 90°. 



and Prof. R. T. Crawford. § The methods by Laves and by 
Crawford are the simplest ones, and the latter will be given here 
in conjunction with the methods for deriving the elements 
P, T, and e. 

In Fig. 7 let the ellipse represent the true orbit and the line 
T'C'T\, its projection upon the plane perpendicular to the line 
of sight. Let the curve APA be the interpolating curve of 
observed distances, obtained in the usual way by plotting the 
distances against the times and drawing the most probable 
smooth curve to represent the plotted points. 

The revolution period may be read directly from this curve 
and the accuracy of its determination will increase with the 
number of observed revolutions. 

In the true orbit, let S be the position of the primary star and 
C, the center of the ellipse. Then the points C and S f on the 
projected orbit are known for T'C {— Ti'C) must be half the 

* Mon. Not. R.A.S., 35, 367, 1875. 

t A.J. 37, 97, 1926. 

X Handbuch der Astrophysik, 6, 338, 1928. 

§ Lick Obs. BuU. 14, 6, 1928. 



THE ORBIT OF A VISUAL BINARY STAR 115 

amplitude of the curve of distances and S' must be the apparent 
position of the primary star. 

The points on the curve of distances which correspond to the 
points P and Pi of the true orbit must be separated by exactly 
half of the revolution period and their distances from the line 
C'CE must be equal in length and of opposite sign. The point 
corresponding to periastron, P, must lie on the same side of this 
line as S' t and on the steeper branch of the curve as the rate 
of change of distance is greater near periastron than near 
apastron. In practice these two points are readily found by 
cutting a rectangular slip of paper to a width equal to half that 
of the period on the adopted scale and sliding it along the curve 
until the edges, kept perpendicular to the line C'CE, cut equal 
ordinates (with respect to C'CE) on the curve; or we may adopt 
Laves's suggestion to "draw the central axis of the graph and 
make a tracing of the time graph of distances, turn it through 
180° about the central time axis and then advance the tracing 
by half the period of the orbit. Of the four points of intersection 
of graph and tracing it is easy to find that pair which is separated 
by half the period." In either of these ways the positions of the 
points P' and Pi' are obtained. 

Since ratios are not altered in the projection we have 

6 = CP' ^ 55 ^ 

To obtain the two remaining elements a and w, Crawford makes 
use of a relation involving the perpendicular from the center 
of the ellipse on a tangent to the ellipse. In the figure let CH 
be such a perpendicular and call its length p. Let co (which is 
here our element u) be the angle the tangent line makes with the 
minor axis. Then, if a and 6 are the major and minor semiaxes, 
respectively, from the properties of the conic it can be shown that 

p2 _ 2 cog 2 w _j_ ^2 gm 2 w 

which can be written 

p2 = a 2 — (a 2 - 6 2 ) sin 2 w = a 2 - a 2 e 2 sin 2 «. (56) 

But, from the figure, 

CK = m = CS cos ct) = ae cos cu 



116 THE BINARY STARS 

Hence 



and 



cos u> — — (57) 



t tft 

sin2a, = 1 -^ < 58 > 



Substituting (58) into (56) and solving for o, we have 



«->£ 



=^ (59) 



p is C'T', and m is C'S', both of which are known. 

The order of solution is, then, Eqs. (55), (59), and (57) which 
give e, a, and «, respectively. 

If the major axis is directed toward the Earth, the points 
C", P', and S' coincide and « = 90°. Equation (55) then becomes 
indeterminate, but at the same time p = b and m = 0. Hence, 
from Eq. (59) we have 



« - Jr^p- (60) 

To determine e, let t be the ratio of the intervals of time 
from elongation to elongation such that t < 1. Then from 
the law of areas we have t equal to the ratio of the area described 
by the radius vector in the true orbit in the shorter interval to the 
area described by the radius vector in the longer interval, or 



^7T06 


— abe 




+ 


e 


}4,irab + abe 


e 


e = 


IT 1 " 

2 1 - 


— T 
f T 







whence 

ir 1 — - T 

(61) 

The time of periastron passage, T, is the middle instant of the 
shorter interval between two consecutive elongation times. 
If the intervals are all equal, r — 1, e = 0, and the orbit is a 
circle, in which case the elements o> and T lose significance. 

When a preliminary set of elements has been derived, improved 
values may be computed by the method of least squares, using 



THE ORBIT OF A VISUAL BINARY STAR 117 

Comstock's expressions for the differential coefficients in the 
equation for distance. 

THE ORBIT OF e EQUULEI 

An excellent example of an orbit whose computation was 
made possible by taking advantage of the special features of 
the observed motion is that of e Equulei, published by Russell 
in 1917.* The apparent orbit of this system is an extremely 
narrow and elongated ellipse. Fortunately the double star was 
discovered by Struve in 1835, when the angular separation 
was only 0"35. In later years the companion moved out to a 
maximum elongation of 1"05 and then in again until, in 1917, 
the pair could not be resolved by any existing telescope. Plotting 
the distances (using mean places) against the times, Russell 
noted that the curve was practically symmetrical with respect 
to the maximum separation point. It follows that the line 
of apsides in the true orbit must be approximately coincident 
with the line of nodes, or that w = 0. Further, the mean of 
the position angles for a few years on either side of the time of 
elongation gives a preliminary value for the angle ft, and the 
elongation time itself is the epoch of apastron passage, which 
may be taken in place of the epoch of periastron as one of the 
orbit elements. It is also apparent that the inclination of the 
orbit is very high and a preliminary value for this element may 
be assumed. This leaves the three elements, a, e, and P, which 
Russell finds as follows: Let 

yx = the maximum elongation distance 

T' = the corresponding epoch (i.e., apastron) 

y" = the distance at any other time t" 
E = the corresponding eccentric anomaly in the true orbit. 

Then we have 

a(l + e) = yi f a(cos E — e) = — y" 
M = E - e cos E, and (t" - T) = 180° - M 

which determine a and P in terms of e. 

Assume values of e and compute a and P, repeating the 
process until those values result which represent the curve of 
the observed distances. 

The preliminary elements obtained by these processes Russell 
corrected differentially, a, e, T and /* from the observed distances, 

* A. J. 30, 123, 1917. 



118 THE BINARY STARS 

i and Q. from the observed angles; to (= 0) being assumed as 
definitely known,* 

SYSTEMS IN WHICH ONE COMPONENT IS INVISIBLE 

Luminosity, Bessel said long ago, is not a necessary attribute 
of stellar mass, and it may happen that one component of a 
double star system is so feebly luminous as not to be visible in 
existing telescopes. If the orbit is one of short period and the 
inclination of its plane sufficiently high, the system may be 
detected by the spectroscope, by the methods to be discussed 
in the following chapter. In other instances the companion's 
presence may be revealed by a periodic variation in the bright 
star's proper motion, the path described by it upon the celestial 
sphere becoming a cycloid instead of the arc of a great circle. 
A system of the latter type is most readily detected when the 
proper motion is large, and it is of course essential that the 
motion be accurately determined. 

Variable proper motion was actually recognized in the stars 
Sirius and Procyon, nearly a century ago, and was explained by 
Bessel as the effect of the attraction of such invisible com- 
panions. Orbits, referring the motion of the bright star to the 
center of gravity of a binary system, were thereupon computed 
for these stars by C. A. F. Peters and A. Auwers. Bessel's 
hypothesis was proven to be correct by the subsequent dis- 
covery of a faint companion to Sirius by Alvan G. Clark (in 
1861), and of a still fainter companion to Procyon by Schaeberle 
(in 1896). The relative orbit of the companion to Sirius has 
been computed from the micrometer measures, and the elements 
are consistent with those determined from the proper motion of 
the bright star. There is no question but that this will also prove 
to be the case in the system of Procyon when the micrometer 
measures permit an independent determination of its orbit. 

Dark companions to fi. Orionis, a Hydrae, and a Virginia 
have also been suspected from supposed irregularities in the 
proper motions, but closer examination of the data has not 
verified the suspicion. Since cases of this kind will probably 
always be very exceptional, the formulas for their investiga- 

* More recent observations indicate the need of revision of Russell's 
elements. 



THE ORBIT OF A VISUAL BINARY STAR 119 

tion will not be considered here. Those who are interested in 
their development are referred to the original memoirs.* 

The presence of invisible companions in several well-known 
double star systems has also been revealed by the observed 
periodic variations in the motion of one of the visible com- 
ponents. In one of these, e Hydrae, the primary star was 
later found to be a very close pair whose components complete 
a revolution in about fifteen years, and Seeligerf has shown 
that the orbital motion in this close pair fully accounts for the 
irregularities observed in the motion of the more distant com- 
panion. Another of these systems, £" Cancri, consists of three 
bright stars, two of which revolve about a common center in a 
period of approximately 60 years, while the third star revolves 
with this binary system in a much larger orbit. Seeliger has 
shown that the irregularities observed in the apparent motion 
of this third star may be explained on the hypothesis that it is 
accompanied by an invisible star, the two revolving about a 
common center in circular orbits with a period of 18 years. 
The system then, would, be a quadruple one. 

Again, Norlund, in the course of his investigation of the orbit of 
£ Ursae Majoris, in 1905, discovered a perturbation of 1.8 years 
period with an amplitude of 0"05. He attributed this to the 
presence of an invisible companion to the brighter component, 
the two bodies revolving in an orbit inclined nearly 90° to the 
plane of projection. Such a companion had, as a matter of 
fact, been discovered by Wright, five years earlier, from the vari- 
able radial velocity of the bright star, but Norlund was unaware 
of the discovery when he announced his result. Van den BosJ 
has made a complete investigation of this triple system, § utilizing 

* Bessel, A. N. 22, 145, 169, 185, 1845. 
Peters, A. N. 32, 1, 17, 33, 49, 1851. 

Auwers, A. N. 63, 273, 1865 and Untersuchungen uber verdnderliche 
Eigenbewegung, 1. Theil, Konigsberg, 1862; 2. Theil, Leipzig, 1868. See 
also A. N. 129, 185, 1892. 

t A. N. 173, 321, 1906. 

t Mem. de I'Acad. Roy. des Sciences et des Lettres de Danemark. Sec. des 
Sci. 8 me Serie, 12, No. 2, 1928. 

§ It is really a quadruple system, for an invisible companion to the fainter 
visual component was discovered in 1918 from spectrograms taken at the 
Lick Observatory. Berman's orbit (Lick Obs. Bull. 15, 109, 1931), however, 
shows that the revolution period of this pair is only 3.98 days. No sensible 
perturbation in the visual orbit can be produced by it. 



120 THE BINARY STARS 

both the micrometric and photographic measures of the bright 
pair and the spectrographs observations of the primary star 
and its invisible companion. There are irregularities in the 
observed motion of 70 Ophiuchi which have been regarded as 
due to the perturbations produced by a third body, but a really 
satisfactory solution of the orbit has not yet been published. 
Finally, we may refer to Comstock's investigation* of the orbital 
motion in the system f Herculis from which he concludes that 
small irregularities in the areal velocity of the bright pair may be 
represented as the effect of an invisible companion to one com- 
ponent, having a periodic time of 18 years and an amplitude less 
than 0"1. Comstock, however, points out that when the sys- 
tematic errors of the observers are determined and allowed for, 
the orbit, without the assumption of a third body, "satisfies the 
observations within the limits of error commonly deemed satis- 
factory." The paper is an excellent example of the method by 
which systematic errors should be investigated in the computa- 
tion of a definitive double star orbit. 

It is probable that the invisible companion in such a system 
as that of f Herculis revolves, like the bright components, in 
an elliptic, rather than a circular orbit; and it is not at all 
improbable that the plane of this orbit is inclined at a greater 
or less angle to the plane of the orbit of the visible system. 
To determine the eccentricity and the inclination, however, 
would greatly complicate the problem and the precision of the 
observational data is not sufficient to warrant such refinements 
unless, as in the case of £ Ursae Majoris, spectrographs observa- 
tions are also available. In practice, it has been found satis- 
factory, in general, to assume that the invisible body moves in a 
circle in the plane of the orbit of the visible stars of the system. 
This assumption leaves but two elements to be determined, the 
period and the radius or semiamplitude, and the formulas for 
these are quite simple. Comstock's formulas for the companion 
in the system of f Herculis, for example, are as follows : 

Let 6, p, represent the polar coordinates of the visible com- 
panion referred to the primary star; \f/, r the corresponding 
coordinates of the center of gravity of the assumed system 
(i.e., the system comprised of the secondary bright star and 
its dark companion) referred to the same origin; and v, a, the 
coordinates of the visible companion referred to the center of 

* A. J. 30, 139, 1917. 



THE ORBIT OF A VISUAL BINARY STAR 121 

mass of itself and its dark companion. Then we shall have 
from the geometrical relations involved, 

p 2 = r 2 _j_ a 2 _|_ 2ar cos (v - f) 

S = ^ + - sin {v - $) (62) 

P 

If we assume that a/r and r/p are quantities whose squares 
are negligibly small, we have by differentiation 



A6 J4> , .Jdv d\f\ . , dr 



Since the assumed system is circular, a and dv/dt are constant 

d\l/ 
quantities, r 2 -rr is also a constant, and a is so small that, in 

the second member of the equation, we may write for ^ and 
p in place of r without sensible error. If, further, for brevity, 
we put d4//dt = K/p 2 and k = dv/dt, the equation takes the 
form 



p 2 ^ = K + a(k P + -J cos(w - 0) - a sin(y - 0)^? (64) 



RECTILINEAR MOTION 

The relative motion in some double stars is apparently recti- 
linear and it is desirable to have criteria which shall enable us 
to decide whether this results from the fact that the orbit is a 
very elongated ellipse, or from the fact that the two stars are 
unrelated and are changing their relative positions by reason 
of the difference in their proper motions. One excellent test, 
which has been applied by Lewis to many of the Struve stars, 
is to plot the path of the companion relative to the primary 
and note whether its motion along that path is uniform or whether 
it becomes more rapid as the distance between it and the primary 
diminishes. In the former case the two stars are independent, 
in the latter, they are physically related. 

A more rigorous test is the one applied, for example, by 
Schlesinger and Alter* to the motion of 61 Cygni. If the motion 
is uniform and in a straight line, the position angles and distances 
of the companion referred to the primary may be represented 
by the equations 

*Publ. Allegheny Obs. 2, 13, 1910. 



122 



THE BINARY STARS 



p 2 = 2 _|_ (t _ y)2 m ! 



m, 



tan (0 - 0) = -£(< - 7) 



(65) 



in which a is the perpendicular distance from the primary, con- 
sidered as fixed, to the path of the companion; <f> is the position 
angle of this perpendicular; T, the time when the com- 
panion was at the foot of the perpendicular, and m, the 
annual relative rectilinear motion of the companion. Approxi- 
mate values for these four quantities may be obtained from 
a plot of the observations and residuals may then be formed 




Fig. 8. — Rectilinear motion. 

by comparing the positions computed from the formulas with 
the observations. If these residuals exhibit no systematic 
character, rectilinear motion may be assumed; if they show a 
systematic course a closer examination is in order to decide 
whether this is due to chance or to orbital motion. In the 
latter case, the indicated curve must be concave to the primary 
and the systematic run of the residuals should be quite uni- 
form. In any event, a least squares solution may be made to 
obtain more precise values for the quantities a, <£, T and m. 
For this purpose, differentiate Eqs. (65) and introduce the 

values sin (0 - </>) = ^-^ — } -, cos (0 -<*>) = - p (see Fig. 8) ; we 



THE ORBIT OF A VISUAL BINARY STAR 123 

thus obtain the equations of condition in the form given by 
Schlesinger and Alter: 



— cos (6 — <j> )Aa — sin (0 — <f> )(t — To) Am 

+ sin (0 — <t>o)moAT + Ap = v p 
+ sin (0 — <£ )Aa — cos (0 — <j> )(t — T )Am 

+ cos (0 — 0o)w o A7 T — pA<f> + pA0 = v B \ 



(66) 



in which the subscript o indicates the preliminary values of the 
elements, Ap and Ad the deviations from the approximate 
straight line, and v p and v e the residuals from the definitive 
values of the elements. 

References 

In addition to the papers cited in the footnotes to the chapter, the student 
of double star orbit methods will find the following of interest: 
Klinkerfues: Uber die Berechnung der Bahnen der Doppelsterne, A.N. 
42, 81, 1855. 

: Allgemeine Methode zur Berechnung von Doppelsternbahnen, 

A.N., 47, 353, 1858. 
Thiele: tJber einen geometrischen Satz zur Berechnung von Doppel- 
sternbahnen, u. s. w., A.N., 52, 39, 1860. 

: Unders0gelse af Oml0bsbevaegelsen i Dobbelstjernesystemet 

7 Virginis, Kj0benhavn, 1866. 

Neue Methode zur Berechnung von Doppelsternbahnen., A.N., 104, 



245, 1883. 

Seeligeb: Untersuchungen iiber die Bewegungsverhaltnisse in dem drei- 
fachen Stern-system f Cancri, Wien, 1881. 

: Fortgesetzte Untersuchungen liber das mehrfache Stern-system f 

Cancri, Miinchen, 1888. 

Schorr: Untersuchungen iiber die Bewegungsverhaltnisse in dem drei- 
fachen Stern-system £ Scorpii, Miinchen, 1889. 

Schwarzschild: Methode zur Bahnbestimmung der Doppelsterne, A.N., 
124, 215, 1890. 

Rambaut: On a Geometrical Method of Finding the Most Probable Appar- 
ent Orbit of a Double Star, Proc. Roy. Dublin Society, 7, 95, 1891. 

Howard: A Graphical Method for Determining the Apparent Orbits of 
Binary Stars, Astronomy and Astrophysics, 13, 425, 1894. 

Hall: The Orbits of Double Stars, A.J., 14, 89, 1895. 

See: Evolution of the Stellar Systems, Vol. 1, 1896. 

Leuschner: On the Universality of the Law of Gravitation, University of 
California Chronicle, 18, No. 2, 1916. 

Andre: Traite d'Astronomie Stellaire, Vol. 2. 

Also the chapters on double star orbits in such works as Klinkerfues- 
Buchholz, Theoretische Astronomie; Bauschinger, Die Bahnbestimmung 
der Himmelskorper; Crossley, Gledhill, and Wilson, A Handbook of 
Double Stars. 



124 THE BINARY STARS 

Dobebck, W.: On the Orbit of £ Bootis, A.N. 214, 89, 1921. 
Comstock, G. C. : On the Determination of Double Star Orbits from Incom- 
plete Data, A.J., 33, 139, 163, 1921. 
Meyermann, B.: Eine neue graphische und eine halbgraphische Methode 

zur Bestimmung von Doppelsternbahnen, A.N. 215, 179, 1922. 

: Zur Bestimmung von Doppelsternbahnen, A.N. 228, 49, 1926. 

Dawson, B. H.: Provisional Elements of the Binary Star h50H with a Note 

on the Method Employed, A.J. 36, 181, 1926. 
Nassau, J. J., and P. D. Wilkins: Graphical Determinations of Orbits of 

Visual Binary Stars, A.J. 38, 56, 1928. 
Henroteau, F. C. : Double and Multiple Stars, Handbuch der Astrophysik, 

Bd. VI, 2 ter Teil, Chapter 4, 1928. 
Parvulesco, C: M6thode nouvelle pour calculer des Orbites d'Etoiles 

doubles, Bull. Lyon Obs. 10, 49, 1928. 
: Contribution a la determination de l'orbite apparente d'une Etoile 

double, d'apres la variation de Tangle de position, Bull. Lyon Obs. 12, 

122, 1930. 
Vahlen, Th.: Doppelsternbahn aus sieben Beobachtungen, A.N. 233, 217, 

1928. 
Kerrich, J. E. : A Method for the Computation of the Orbital Elements for 

Certain Binary Stars, Union Obs. Circ. 82, 123, 1930. 
Volet, Ch.: Application de la methode des moindres carrees au calcul des 

orbites d'&oiles doubles, C.R. 192, 482, 1931. 
: Methode pour le Calcul des Orbites d'Etoiles Doubles Visuelles, 

Application a l'Orbite du Compagnon de Sirius, Butt. Astron., Ser. 2, 

7, 13, 1931. 
Russell, H. N.: A Rapid Method for Determining Visual Binary Orbits, 

Mon. Not. R.A.S. 93, 599, 1933. 
De Sitter, W. : On the Solution of Normal Equations, Ann. Cape Obs. 12, 

Pt. 1. 160-173. 



CHAPTER V 

THE RADIAL VELOCITY OF A STAR 

By J. H. Moore 

The observations treated in the preceding chapters concern 
only that part of the star's actual motion in space, which appears 
as change of position in a plane perpendicular to the line joining 
the observer and star. Of the component directed along the 
line of sight, called the star's radial motion, the telescope alone 
gives no indication. In fact, the possibility of detecting radial 
motion was recognized less than a century ago, and the methods 
of its measurement belong distinctly to another and newer 
branch of astronomy, known as astrophysics. Moreover, 
observations of the rate of change of position of a star on the 
celestial sphere can be translated into linear units, such as 
kilometers per second, only if the star's parallax is known, while 
measures of radial velocity by the method to be described, 
are expressed directly in kilometers per second and are inde- 
pendent of the star's distance. 

The determination of the radial velocity of a light source, 
such as a star, is made possible by two well-known properties 
of light; viz., that it is propagated as a wave motion, and with 
a definite and finite velocity. We are not concerned with 
the properties of the hypothetical medium, called the ether, 
in which these waves move, nor with the nature of the dis- 
turbance in the ether, whether it be mechanical or electromag- 
netic. For our purpose it is sufficient to know that in this 
medium, or in interstellar space, the velocity of light is about 
299,796 km/sec. and that the well-known laws of wave motion 
hold for light waves. 

In 1842, Christian Doppler called attention to an effect 
upon the apparent length of a wave which should result from 
a relative motion of the source of the waves and the observer. 
This result was independently reached and further developed, 
especially with reference to light waves, some six years later 
by the great French physicist, Fizeau. According to the 

125 



126 THE BINARY STARS 

Doppler-Fizeau principle, when the relative motion of the light 
source and the observer is such, that the distance between the 
two is increasing or decreasing, the length of the waves received 
by the observer will be longer or shorter, respectively, than the 
normal length of these waves. 

It is readily shown that the change in wave length is directly 
proportional to the normal length of the wave and to the ratio 
of the relative velocity of source and observer to the velocity 
of propagation of the waves. Moreover, for light waves the 
change is the same whether the source, or observer, or both are 
moving and depends only upon the relative velocity of the two. 

Let us denote by v the relative radial velocity in kilometers 
per second of a star and observer, where v is considered positive 
when the distance between the two is increasing and negative 
when this distance is decreasing. Call X' the wave length of a 
monochromatic ray reaching the observer, whose normal wave 
length, as emitted by the star, is X. 

Then from the Doppler-Fizeau principle, X 7 — X:X: :v: 299,796; 
or X' — X = Xv/299,796 (if v is +, X' is greater than X) ; or, writing 
AX for the change in wave length (X' — X), we have for the 
relative radial velocity of star and observer 

299,796AX „.„ 

v = ^ (1)* 

The determination of the radial velocity of a star rests then 
upon a knowledge of the velocity of light and of the wave 
lengths of certain definite rays emitted by a source at rest, and 
the measurement of the apparent wave lengths of those same 
rays received from a star. In short, the problem reduces to one 
of measuring AX with the greatest possible precision. For this 
purpose the micrometer, with which we have become familiar, 
is replaced by the spectroscope. This wonderful instrument 
originating in the physical laboratory has developed a whole 
new science, spectroscopy, with an extensive and technical 
literature of its own. In this chapter we shall only call atten- 
tion to some of the elementary principles of spectroscopic 
analysis and give a very brief survey of the spectrographic method 

* It may be shown from the theory of relativity that this formula holds 
for the relative velocity of source and observer where this is small in com- 
parison with the velocity of light. This condition is fulfilled by the stellar 
light sources with which we are here concerned. 



THE RADIAL VELOCITY OF A STAR 127 

as applied to the determination of stellar radial velocities. 
The student who wishes to pursue the subject further will 
find a list of references to extended treatment of the various 
topics at the end of this chapter. 

Since stellar light sources are very faint in comparison with 
those available in the laboratory, it is necessary to employ for 
this special problem the spectroscope which is the least wasteful 
of light. For this reason the prism spectroscope is the only 
one of the various laboratory forms which is at present generally 
applicable to stellar spectroscopy* and therefore we limit our 
discussion to this particular type. 

The essential parts of a laboratory spectroscope and their 
principal functions are briefly as follows: Light from the 
source to be studied is brought to a focus by a condensing lens 
on the narrow slit of the spectroscope. After passing through 
the slit, the rays are rendered parallel by an achromatic con- 
verging lens, called the collimator lens. The rays then strike 
a glass prism, placed with its apex parallel to the length of 
the slit, by which they are bent from their original direction. 
It is here that we obtain the separation of the rays, since the 
amount by which each ray is deviated by the prism is a func- 
tion of its wave length. The direction of the long red waves 
is changed the least, while the shorter violet ones suffer the 
greatest deviation. After each set of rays is collected and 
brought to its corresponding focus by a second achromatic 
converging lens, we shall have an orderly array of images of 
the slit, each image formed by light of a definite wave length. 
Such a series of images is called a spectrum of the source. The 
spectrum may be viewed with an ordinary eyepiece, or the 
second lens may be used as a camera lens, and the spectrum 
be recorded on a photographic plate placed in its focal plane. 
When the spectroscope is employed photographically, as it is 
in practically all stellar work, it is called a spectrograph, and the 
photograph obtained with it is a spectrogram. 

If the slit is made extremely narrow, there will be very little 
overlapping of the images and the spectrum is then said to be 

* In recent years, grating spectrographs of high dispersion have been 
successfully employed with large telescopes for studies of the spectra of the 
brighter stars. This instrument has also proved more efficient for the 
investigation of the spectra of the red stars on account of the greater dis- 
persion given by the grating in the region of longer wave length. 



128 THE BINARY STARS 

pure. It can be shown that the purest spectrum is obtained 
when the incident rays fall upon the prism at such an angle 
that they will be least deviated from their original direction 
by the prism. It is well known that this position of minimum 
deviation is also the one of maximum light transmission by the 
prism; and it has the further advantage that any accidental 
displacement of the prism produces the minimum displace- 
ment of the spectrum line. The prism or prisms of stellar 
spectrographs are therefore always set at the angle of mini- 
mum deviation for the approximate center of the region of 
spectrum to be studied. 

Attention was called in an earlier chapter to two factors 
which define the optical efficiency of a telescope for the sepa- 
ration of close double stars, viz.: (a) the resolving power of 
the objective, (6) the magnification or linear distance between 
the two images at the focus of the objective. These same 
factors form a convenient basis for the comparison of the resolving 
powers of two spectrographs. Here, however, we are con- 
cerned with the separation of two images of the slit formed 
by light of different wave lengths. The resolving power of 
a spectrograph is, therefore, denned as the minimum difference 
of wave length between two lines for which the lines will just be 
separated. It is a function of the width of slit, the wave length, 
and the difference between the maximum and minimum lengths 
of path of the rays in the prism. The magnification, called 
the linear dispersion of the spectrograph, is expressed, as the 
number of wave length units per unit length of spectrum and 
depends upon the wave length of the ray, the optical constants 
for the prism system and the focal length of the camera lens. 

When the slit of a spectroscope is illuminated by the light 
from an incandescent solid, such as the filament of an incan- 
descent lamp, or from an incandescent gas under high pressure, 
the spectrum consists of an unbroken band of color; that is, a 
continuous spectrum. An incandescent gas or vapor under 
low pressure gives a spectrum consisting of isolated bright 
line images of the slit, a bright line spectrum, the bright lines 
indicating that radiations of certain definite wave length are 
emitted by the gas. Each chemical element, in the gaseous 
state, when rendered luminous in the electric arc, electric spark, 
flame, or vacuum tube, gives its own set of bright lines which 
are characteristic of this element alone and whose wave lengths 



THE RADIAL VELOCITY OF A STAR 129 

remain constant for a source at rest under the same conditions 
of temperature, pressure, etc. 

An incandescent gas has the property not only of radiating 
light of certain definite wave lengths, but also of absorbing, 
from white light passing through it, the rays of precisely those 
same wave lengths. If the temperature of the incandescent 
gas is lower than that of the source behind it, the continuous 
spectrum will be crossed by relatively dark lines whose posi- 
tions agree exactly with the bright line spectrum characteristic 
of the gas. This relation existing between the emission and 
absorption of a gas is known as Kirchhoff's law, and the type 
of spectrum described is termed an absorption spectrum. 

The three principles just stated obviously lead to a simple 
and direct method of analyzing the chemical constituents of 
gaseous light sources and of furnishing information as to their 
physical conditions. Nebulae of a certain class, for example, 
give bright line spectra, indicating that they are masses of 
luminous and extremely rarefied gases. Most of the stars, 
including our own Sun, give absorption spectra, showing that 
the light emitted by a central glowing core has passed through 
a surrounding atmosphere of cooler vapors. The presence in 
the atmospheres of the Sun and stars of most of the known 
chemical elements has been recognized from the lines in the 
spectra of these objects. In addition, there occur in these 
spectra many lines, which have not yet been identified with 
those of any known element. 

The length of the light wave for each line is such a minute 
fraction of a millimeter that spectroscopists have adopted as 
the unit of wave length, the Angstrom, equal to 0.0000001 mm. 
for which A is the symbol. Thus the wave length of the hydrogen 
radiation in the violet is 0.0004340 mm. or 4340A. 

Measures of the wave lengths of the lines in a star's spec- 
trum secured with the prism spectrograph, are readily effected 
by a comparison of the positions of the stellar lines with those 
from a source the wave lengths of whose lines are known. To 
accomplish this the light from a suitable source (for example, 
the iron arc) is made to pass over very nearly the same path 
in the spectrograph as that over which the star's light travels, 
and the spectrum of this source, termed the comparison spectrum, 
is recorded on each side of the star spectrum. 



130 THE BINARY STARS 

When the spectra of a number of stars are examined, it is 
found that they exhibit a great variety in the number and 
character of their lines. From an examination of several 
hundred stars by means of a visual spectroscope, Secchi, about 
1866-1867, was able to arrange their spectra under four types. 
While exhibiting very well the most prominent characteristics 
of stellar spectra, his system is insufficient for portraying the 
finer gradations, which the photographic method has brought 
to light. The classification now in general use among astro- 
physicists was formulated by Prof. Pickering, Miss Maury, 
and Miss Cannon from the very extensive photographic survey 
of stellar spectra made at the Harvard College Observatory and 
at the Harvard station at Arequipa, Peru. It is based upon the 
observed fact that certain groups of lines have a common behav- 
ior. They make their appearance and increase or decrease in 
intensity at the same time, so that a more or less orderly sequence 
of development from one type of spectrum to another is indicated. 

A condensed outline of this system will serve to indicate its 
chief features. The main divisions are represented by the 
letters, O, B, A, F, G, K, M (R, N, S). Classes B to M, in the 
order given, form a continuous sequence, and types intermediate 
between the main ones are indicated by numbers 1 to 9 inclusive. 
Class O undoubtedly precedes Class B, but its subdivisions are 
still provisional and are indicated by small letters a to e. Classes 
R and N appear to form a branch from the main sequence begin- 
ning at Class K, while Class S seems to be still another such 
offshoot. 

In Class O, subdivisions Oa to Oe show faint continuous spectra 
upon which are superposed bright bands. The lines of hydrogen 
and helium are bright in the beginning of the class but dark in the 
later subdivisions. Characteristic lines of the class are those of 
ionized helium,* and doubly and trebly ionized oxygen, nitrogen, 

* On the basis of the atomic model that considers the atom to consist of a 
positively charged nucleus about which revolve negatively charged units 
called electrons, it is possible to picture the manner in which the atoms 
radiate energy of definite frequencies corresponding to the different spectral 
lines. Each atom in the neutral state has a definite number of electrons — 
hydrogen 1, helium 2, lithium 3, etc. — and to the atom of each element in 
this state there corresponds a characteristic spectrum. If one electron is 
removed from the atom, the latter is said to be singly ionized, if two are 
removed, doubly ionized. Thus the atoms of ionized helium and doubly 
ionized lithium have only one electron and give entirely different spectra 
from those given by their neutral atoms. In the laboratory it is found 



THE RADIAL VELOCITY OF A STAR 131 

and silicon. Prominent features of Class B are the dark lines of 
hydrogen and neutral helium. Near the end of the class the 
helium lines weaken and they are absent in AO, while the lines 
given by the ionized atoms of the metals magnesium, calcium, 
iron, etc., begin to appear. The hydrogen lines reach their 
maximum intensity early in Class A and steadily decrease in 
strength through the remainder of the spectral sequence. The 
lines of ionized calcium, H and K, and those of the metals 
increase in prominence through this class and the subdivisions 
of Class F. In Class G, which includes stars whose spectra 
closely resemble that of the Sun, the H and K lines and the 
numerous metallic lines are conspicuous features, whereas 
hydrogen has become less prominent. Class K spectra show a 
further weakening of the enhanced lines of the metals and 
strengthening of the arc lines, especially those which appear in 
the laboratory at lower temperatures. In Class M these low- 
temperature lines are still further strengthened as the high-tem- 
perature lines decrease in intensity. This class is characterized 
by the absorption bands of titanium oxide which first make 
their appearance at K5 and increase in intensity through the 
subdivisions of Class M. Stars of Class Me show, in addition, 
bright hydrogen lines. To classes R and N belong stars whose 
spectra of metallic lines are similar to those of M but which are 
particularly characterized by bands of carbon and cyanogen. 
Spectra of Class S are likewise similar to M in the strengthening 
of the low temperature lines of the metals, but in this class the 
bands of zirconium oxide are present in addition to those of 
titanium oxide. 

Stars of classes O and B are bluish white in color; those of 
Class A, white; of F and G, yellow; of K, orange; of M, R, and S, 
red; while the N stars are a deep red. 

In Plate III are reproduced four stellar spectrograms secured 
with the three-prism spectrograph of the D. 0. Mills Expedi- 
tion, at Santiago, Chile, which illustrate the different appear- 
ance of the spectra in the blue-violet region of classes B8, F, 
G, and K5. On all of the spectrograms the bright line spectrum 

that the lines from the ionized atom are stronger in the spark (enhanced) 
as compared with their intensities in the electric arc, while in the latter the 
lines of the neutral atom are stronger. The lines of the ionized atom are 
therefore frequently referred to as enhanced and those of the neutral atom 
as arc lines. 



132 



THE BINARY STARS 



of the iron arc was photographed above and below the star 
spectrum. The spectrum of v* Eridani (Figure a) of Class B8, 
shows only the hydrogen line H7 (4340. 4 77 A) and the magnesium 




i 
si 

"1 IHBIB 

line (4481. 228 A), as the very faint metallic lines, some of which 
appear on the original negative, are lost in the process of reproduc- 
tion. This star is a spectroscopic binary, and the spectra of 
both stars are visible, so that each of the two lines mentioned 



THE RADIAL VELOCITY OF A STAR 133 

above is double. The strengthening of the metallic lines and the 
decrease in intensity of H7 are shown in the spectrum of a Carinae 
of Class F (Fig. b), while in the solar spectrum (Fig. c), of Class G, 
and in that of a 2 Centauri (Fig. d), of Class K5, a further decrease 
in Hy, the disappearance of 4481 A and a considerable increase 
in the number and strength of the absorption lines of other 
elements are noticeable. 

The four spectrograms illustrate also the displacement of 
the lines in star spectra as effects of motion in the line of sight. 
The iron lines in the solar spectrum are practically coincident 
with the corresponding lines of the iron arc, since the relative 
radial velocity of the Sun and the observer is very small. The 
iron lines in the spectrum of a Carinae are clearly displaced 
from their normal positions, as given by the lines of the com- 
parison spectrum. This displacement is toward the red end 
of the spectrum, and corresponds, therefore, to an increase in 
the wave lengths of the star lines. Interpreted on the Doppler- 
Fizeau principle, this change is produced by a recession of the 
star with respect to the Earth at the rate of +18.0 km/sec. 
In the case of a 2 Centauri, the displacement of the lines is 
toward the violet and corresponds to a velocity of approach of 
—35.0 km/sec. As an example of the Doppler-Fizeau effect, 
the spectrogram of the spectroscopic binary u 4 Eridani, is perhaps 
the most striking. The two stars revolve about their common 
center of mass in a period of 5.01 days, as shown by an extended 
series of plates similar to this one. Due to their orbital motion, 
the velocity of each star in the line of sight is continually chang- 
ing, giving rise to a continuous variation in the separation of the 
lines of the two spectra. The spectrogram reproduced here was 
taken at the time of maximum velocity of approach of one, and 
the corresponding velocity of recession of the other component. 
It shows, therefore, the maximum separation of the lines of 
the two spectra. The relative radial velocity of the two stars 
was 132 km/sec. Obviously, the lines of the two spectra will be 
coincident when the motion of the two components is across the 
line of sight, which occurs at intervals of 2.5 days. 

It is well known that the wave lengths of spectral lines are 
affected by other causes than that arising from radial motion 
of the source. For example, it is found that an increase in 
pressure of the emitting or absorbing vapor will in general shift 
the lines toward the red. This effect, even with considerable 



134 THE BINARY STARS 

pressures, is small and is moreover not the same for all lines. 
Of the many conditions which displace spectrum lines, radial 
motion is the only one of which measures of stellar spectra 
have furnished reliable evidence. * 

Displacements of the stellar lines, with reference to those of 
the comparison spectrum, may arise wholly or in part from 
causes which are purely instrumental. Thus, if the starlight 
and the artificial light do not pass over equivalent paths in 
the spectrograph, or if a change in the relative positions of the 
parts of the instrument occurs between the times of photograph- 
ing the stellar and the reference spectra, a relative displacement 
of the lines of the two spectra will result. The first-named 
source of error is an optical condition, to be met for all spectro- 
scopic measures, that is easily satisfied. With the conditions 
of a fixed mounting and approximately constant temperature, 
under which the spectrograph is used in the laboratory, the 
second source of error need not be considered. When, however, 
the spectrograph is applied to stellar observation, it is necessary, 
in order to avoid undue loss of light, to mount it on a moving 
telescope, and hence to subject the instrument to the varying 
component of gravity and the changing temperature of a well- 
ventilated dome. The spectrograph must be so designed and 
constructed that it will be free from appreciable differential 
flexure in any two positions of the telescope, and provision must 
be made against the disturbing effects of temperature changes 
in the prisms and the metal parts of the instrument. Further, 
in addition to the obvious requirement that the prisms and lenses 
shall give good definition, they must be so chosen and arranged 
as to give satisfactory resolving power with efficiency in light 
transmission. 

The earlier determinations of stellar radial velocities were 
made entirely by the visual method. Although made by such 
skilled observers as Huggins, Vogel, and others, the errors of 
observation, except for a very few of the brightest stars, often 
exceeded the quantities to be measured. After the introduc- 
tion of the photographic method of studying stellar spectra, 
Vogel and Scheiner, at Potsdam, and later Belopolsky, at 
Pulkowa, were able to measure the radial velocities of the 

* A possible exception is the lengthening of the light waves in a strong 
gravitational field, evidence of which has been found in the spectrum of the 
companion of Sirius. 



THE RADIAL VELOCITY OF A STAR 135 

brightest stars with an average probable error of ±2.6 km/sec. 
In 1895-1896 the problem was attacked by Campbell, who 
employed a specially designed stellar spectrograph — the Mills 
Spectrograph — in conjunction with the 36-in. refractor of 
the Lick Observatory. For the brighter stars, the probable 
error of his measures was about ±0.5 km and for bright stars 
whose spectra contain the best lines, the probable error was 
reduced to ±0.25 km. Many improvements in stellar spectro- 
graphs have, of course, been made in the succeeding years, 
but the standard of precision, set by his measures nearly 40 years 
ago, represents that attained today for the same stars. The 
advances which have been made in this time relate more to the 
increased accuracy of the results for fainter stars. 

Now this remarkable advance in the precision of the meas- 
ures made by Campbell was due not to the use of a great tele- 
scope but to the fact that his spectrograph was designed in 
accordance with the important requirements mentioned in the 
foregoing — excellence of definition and maximum light trans- 
mission, rigidity, and temperature control of the spectrograph — 
and to improved methods of measuring and reducing the 
spectrograms. 

In order to understand more clearly the manner in which 
the optical and mechanical requirements are met in practice, 
a detailed description will be given of a modern spectrograph 
which was designed to have maximum efficiency for the par- 
ticular problem of determining stellar radial velocities. A 
view of the new Mills spectrograph attached to the 36-in. 
refractor of the Lick Observatory is presented in Plate IV. 
The essential parts of this instrument are. the same as those 
described for the simple laboratory spectrograph; namely, 
the slit, collimator lens, prism, and camera lens, except that 
here three 60° prisms of flint glass are employed. The prisms, 
set at minimum deviation for 4500A, produce a deviation 
of this ray of 176°. A rectangular box constructed of saw-steel 
plates, to which are connected respectively the slit mechanism, 
the prism box, and the plate holder, by three light steel castings, 
forms the main body of the spectrograph. In the casting 
to which the prism box is attached are mounted the collimator 
and camera lenses, both of which are achromatic for the region 
of 4500A. The spectrograph has an entirely new form of sup- 
port, designed by Campbell, to incorporate the suggestion made 



136 



THE BINARY STARS 



by Wright, that such an instrument should be supported near 
its two ends, like a bridge truss or beam, in order to give minimum 
flexure. The support is a frame work of T-bars extending down 
from the telescope, the form and arrangement of which are such 
as to hold the instrument rigidly in the line of collimation of the 




Plate IV. — The Mills spectrograph of the Lick Observatory. 

large telescope. The lower support is a bar passing through a 
rectangular opening in the casting carrying the prism box. 
This bar is pivoted at the center of the casting and connected at 
its two ends to the supporting frame. The upper support 
consists of a cylindrical ring firmly attached to the frame work. 
In this cylinder fits a spherical flange of the spectrograph casting, 
the two forming a universal joint. Any strains originating 



THE RADIAL VELOCITY OF A STAR 137 

in the supporting frame cannot, with this form of mounting, 
be communicated to the spectrograph. Careful tests of this 
instrument and of the spectrograph of the D. 0. Mills Expedition 
to Chile, which has the same form of mounting, show that the 
effects of differential flexure have been eliminated. This method 
of support permits, further, of a very convenient mode of moving 
the spectrograph as a whole in order to bring the slit into the focal 
plane of the large telescope, since it is only necessary to provide 
sliding connections on the frame, for the lower support. 

Nearly all modern stellar spectrographs are provided with 
reflecting slit plates inclined at a small angle to the collimation 
axis, which enable the observer to view the star image directly 
on the slit. This is accomplished through the aid of a total- 
reflection prism, placed above the slit and outside of the cone 
of rays from the telescope objective, which receives the light 
from the slit and sends it to the guiding eyepiece. By placing 
the slit parallel to the celestial equator, small errors of the 
driving clock cause the star image to move along the slit, which 
is desirable in order to obtain width of spectrum. Constant 
and careful guiding is necessary to insure that the star's image 
be kept exactly on the slit and that its motion along the slit be such 
as to give a uniform exposure. 

With a prism spectrograph and a straight slit the spectrum 
lines are curved. The amount of the curvature depends upon 
the optical constants of the instrument and the wave length 
of the line. This source of trouble in measuring the spectro- 
grams may be eliminated for a short range of spectrum by 
employing a slit of the proper curvature to make the spectrum 
lines straight. Both three-prism instruments referred to above 
are provided with curved slits. 

As a source for the comparison spectrum, it is necessary to 
select one giving a number of well-distributed lines in the part 
of the spectrum to be studied. For example, for the new 
Mills spectrograph in which the region 4400 to 4600A is utilized, 
the spark spectrum of titanium is used. In the southern instru- 
ment, arranged for the region 4200 to 4500A, the comparison 
source is the iron arc. 

In order to eliminate the effects of any possible change in 
the instrument during an exposure on the star, several impres- 
sions of the comparison spectrum are made at regular inter- 
vals. This is accomplished very conveniently and without 



138 THE BINARY STARS 

danger of changing the adjustment of the comparison apparatus 
by a simple device due to Wright. Two small total-reflection 
prisms are placed just above the slit, so that their adjoining 
edges define the length of the slit. Two light sources are then 
so arranged that the beam of each is brought to a focus on the 
slit by a small condensing lens after total reflection in its respec- 
tive prism. 

The optical parts of the spectrograph should, of course, be 
mounted so that they cannot move, but care must be taken 
that they are not cramped. This caution is especially pertinent 
with regard to the large prisms. In the Mills spectrographs 
the prisms rest upon hard-rubber blocks and are firmly clamped 
to one of the side plates of the prism box by light steel springs 
which press against their upper surface. Small hard-rubber 
stops prevent lateral motion of the prisms. 

In order to prevent the effects of changing temperature, the 
principal parts of the spectrograph are surrounded by a light 
wooden box, lined with felt. Over the felt surface are strung 
a number of turns of resistance wire. The regulation of the 
heating current is effected by means of a very sensitive mer- 
cury-in-glass thermostat by which the temperature inside of 
the prism box is held constant during the night's work to within 
a few hundredths of a degree centigrade. 

The function of the telescope objective, for observations of 
stellar spectra, is that of a condensing lens and the brightness 
of the point image in the focal plane is directly proportional to 
the area of the lens and its transmission factor. If we had 
perfect seeing, we should receive in the slit of the spectro- 
graph, with the widths generally employed, about 90 per cent 
of the light in the star image. Due to atmospheric disturbances 
the image of a star under average conditions of seeing, is a 
circular tremor disk whose diameter is four or five times the 
width of the slit, so that the brightness of the spectrum is not 
proportional to the area of the objective but more nearly to its 
diameter. For example, the relative intensities of stellar spectra 
obtained with the same spectrograph respectively upon the 
36-in. and 12-in. refractors of the Lick Observatory would be 
(allowing for the difference of transmission of the two), about 
as two to one, since, for the photographic rays, the loss of light is 
for the former about 50 per cent and for the latter about 25 per 
cent. When a visual refractor is used for spectroscopic work, 



THE RADIAL VELOCITY OF A STAR 139 

it is necessary to render it achromatic for the photographic 
rays. This is accomplished for the 36-in. refractor by a correct- 
ing lens of 2j^-in. aperture placed one meter inside the visual 
focus of the telescope. This lens introduces an additional 
loss of light of fully 10 per cent. 

Since a silver-on-glass mirror has, under the best conditions, 
a high reflecting power, and since it is also free from chromatic 
aberration, it would seem that the reflector should be the more 
efficient telescope to use in connection with a stellar spectro- 
graph. The reflector, however, possesses its own disadvan- 
tages, one of which is that it is very sensitive to changes of 
temperature. Our experience with the 36-in. refractor at 
Mount Hamilton and the 373^-in. reflector in Chile, when 
used with high dispersion spectrographs, indicates that the 
relative light efficiency of the two is about equal in the region of 
H7. For apertures up to 36 in. one is inclined to favor the 
refractor for high dispersion work, while for low dispersion, 
where considerable extent of spectrum is desired, the reflector is, 
of course, preferable. 

The focal lengths of both refracting and reflecting telescopes 
vary with change in temperature of the lens or mirror. It is, 
therefore, necessary before beginning the night's work, and, 
with the reflector, frequently during the night, to bring the 
slit into the focal plane of the telescope, which, as noted above, 
is effected by moving the spectrograph as a whole in the line 
of collimation of the instrument. 

It is well known that all high dispersion spectrographs are 
very wasteful of light, though to what extent is perhaps not 
always appreciated. When stellar spectrographs of three- 
prism dispersion are used in conjunction with large refractors 
or reflectors, the combined instrument delivers to the photo- 
graphic plate probably less than 2 per cent of the light incident 
upon the telescope objective. Half of the light is lost, as we 
have noted, before it reaches the slit. The remaining losses 
occur at the slit, in the prisms and in the collimator and camera 
lenses of the spectrograph. In order to avoid unnecessary 
losses of light, the obvious conditions must be satisfied, that 
the angular apertures of the collimator lens and object glass are 
the same, and that the prisms and camera lens are of sufficient 
aperture to admit the full beam from the collimator. The 
most serious losses occur at the narrow slit and in the prism 



140 THE BINARY STARS 

train. Indeed, one of the most important factors in the design 
of stellar spectrographs, for maximum light efficiency, is the 
proper balancing of these two conflicting elements, the trans- 
mission at the slit and the transmission of the prisms. Thus, 
in the new Mills spectrograph, by using a collimator of slightly 
greater focal length than the present one (28^ in.) with cor- 
responding increase in aperture of the lens and prisms, a wider 
slit could be employed and still maintain the present purity of 
spectrum. After allowing for the increased absorption of the 
prism train, there would remain a small gain in light transmitted. 
Although theoretically possible, this gain would probably be 
more than offset by the inferior definition of the larger prisms and 
the added difficulty of eliminating flexure. It is necessary here, 
as at so many points in the spectrograph, to sacrifice a little 
in order to gain more elsewhere. In fact, the most efficient 
design of spectrograph may be described as the one in which 
the wisest compromises have been made between the various 
conflicting interests. 

The decision as to the resolution and dispersion to be employed 
is governed by several considerations : the type of stellar spectrum 
to be studied, the size of the telescope at one's disposal, and 
the brightness of the source whose spectrum can be photographed 
with reasonable exposure times. With the spectrograph here 
described two lines in the region of 4500A whose wave lengths 
differ 0.2A are resolved, while the linear dispersion for 4500A 
is 1 mm = 11 A. In order to obtain a spectrogram of suitable 
density of a star whose photographic magnitude is 5.0, an 
exposure time of an hour and a half is required. For stars of 
photographic magnitudes 6.0 to 6.5 the width of the slit is 
increased, thus sacrificing to some degree the purity of spectrum, 
but not enough to interfere seriously with the accuracy of the 
measures. In the case of early- type stars whose spectra contain 
single lines, the question of resolution is not important, and 
where these lines are also broad, it is preferable to employ lower 
dispersion. The adjustments of the various parts of the spectro- 
graph call for continual attention. It is necessary that the 
instrument be placed with its axis of collimation accurately 
in that of the large telescope and frequent tests should be made 
to be sure that it remains so. The comparison source must be 
adjusted so that its light follows very nearly the same path as the 
starlight in the spectrograph. Care must be exercised at 



THE RADIAL VELOCITY OF A STAR 141 

every point in the process of obtaining and measuring the 
spectrogram. 

THE MEASUREMENT AND REDUCTION OF SPECTROGRAMS 

For the measurement of spectrograms any one of the usual 
forms of laboratory measuring microscopes will suffice. This 
is merely a microscope on the stand of which is mounted a 
carriage, movable by an accurate micrometer screw, in a direction 
at right angles to that of the microscope axis. 

In order to fix ideas, we shall assume that it is required to 
measure and reduce a spectrogram of at Centauri, similar to the 
one whose positive is reproduced in Plate III. The spectrogram 
is first clamped on the carriage of the microscope, and the usual 
adjustments of focus and alignment of the plate are made. 
Great care should be taken that the illumination of the field 
of the microscope is uniform. Beginning with the comparison 
line 4250A, settings are made continuously along the plate on 
good star lines and comparison lines as they chance to occur. 
The plate is then reversed and the settings are repeated. It has 
been shown by several investigators that the effects of errors 
due to personal equation are practically eliminated by taking 
the mean of the measures in the two positions. In tne reversal 
of the plate the spectrum is also inverted, which may so change 
the appearance of the lines as to interfere with the elimination 
of personal equation. Especially is this true if the lines are 
curved. The effects of accidental errors in setting are reduced 
by employing a number of lines. 

The table on page 143 contains the data of the measure and 
reduction of this plate. Column I gives the wave lengths in 
terms of the International Angstrom (I.A.) of the lines in the 
iron comparison and the normal wave lengths of the star lines, 
taken from the The Revision of Rowland's Preliminary Table 
of Solar Wave Lengths. In columns IV and V are recorded 
respectively the settings on the comparison and star lines (in 
revolutions of the micrometer screw). The displacements 
of the iron lines in the star are evidently given directly in amount 
and sign by the differences, star minus comparison, and these 
are entered at once in column VII (Displ.). We cannot enter 
the displacements for the other star lines until the normal 
positions of these lines have been obtained from those of the 



142 THE BINARY STARS 

iron comparison, by interpolation. This is effected in the 
following manner: 

A smooth curve drawn by plotting, for the comparison lines, 
the reading on each line and its corresponding wave length, 
respectively as ordinates and abscissae, will evidently repre- 
sent for this spectrogram the relation existing between wave 
length and micrometer readings. From this curve — called a 
dispersion curve — either the zero readings or the observed 
wave length of the stellar lines could be obtained. This curve 
was found by Cornu and later by Hartmann to be nearly of 
the form of an equilateral hyperbola so that it is approximately 
represented by the equation 

X — X = z r- (2) 

A — Ao 

where x is the micrometer reading on a line whose wave length 
is X and Xo, x , and c are constants. Since it is not practicable 
to plot the dispersion curve, the Cornu-Hartmann formula 
furnishes a very convenient means of obtaining it. The values 
of the three constants are determined from three equations 
formed by substituting the micrometer readings and wave 
lengths of three lines, selected, one at each end of the region 
of spectrum and the other near the middle. Micrometer read- 
ings of all other comparison and star lines are then computed 
from the formula. The departure of this computed curve from 
the true dispersion curve is furnished by a plot of the differ- 
ences between the observed and calculated readings of the 
comparison lines. The computed normal positions of the star 
lines are then corrected for the difference between the com- 
puted and observed dispersion curve. The decimal portions of 
the results would be entered in column VI (Sup'd).* 

As before, the difference, star line minus zero line, gives the 
displacement in revolutions of the screw. In order to express 
this as AX, that is in units of wave length, it is necessary to 
know r, the number of Angstrom units in one revolution of the 
screw. The value of r for any point in the spectrum is evi- 
dently the slope of the dispersion curve at that point, and is 
equal to (X — X )/(x — x ). Finally, in accordance with the 

* The figures actually entered in this column in the example were obtained 
by a different method of reduction which is explained in the paragraphs 
following. 



THE RADIAL VELOCITY OF A STAR 



143 



* az Centauri ft. 




* Plate No. 3791 III 


ol4 h 32.8 m 


Date 1911 Feb. 27 












X 


Table 


Oo- 
Ta 


Comp. 


* 


Sup'd. 


Displ. 


TVs 


v, 


4250.132 


54.886 





54.886 


54.758 




-0.128 


319 


-40.8 


4250.799 


55.031 


3 


55.034 


54.909 




-0.125 


320 


-40.0 


4282.413 


61.819 


13 


61.832 


61.710 




-0.122 


335 


-40.9 


4283.016 


61.944 






61.831 


958 


-0.127 


335 


-42.5 


4294.147 


64.250 


16 


64.266 


64.140 




-0.126 


338 


-42.6 


4299.252 


65.295 


20 


65.315 


65.190 




-0.125 


340 


-42.5 


4312.877 


68.039 






67.944 


061 


-0.117 


349 


-40.8 


4313.633 


68.190 






68.090 


212 


-0.122 


349 


-42.6 


4318.660 


69.185 






69.105 


220 


-0.115 


352 


-40.5 


4325.000 


70.431 






70.355 


469 


-0.114 


356 


-40.6 


4325.777 


70.584 


40 


70.624 


70.502 




-0.122 


356 


-43.4 


4327.919 


71.001 






70.928 


041 


-0.113 


357 


-40.3 


4337.057 


72.767 


43 


72.810 


72.692 




-0.118 


360 


-42.5 


4340.477 


73.421 






73.350 


467 


-0.117 


362 


-42.4 


4359.625 


77.027 






76.970 


082 


-0.112 


372 


-41.7 


4369.781 


78.896 






78.844 


957 


-0.113 


376 


-42.5 


4375.946 


80.018 






79.972 


083 


-0.111 


378 


-42.0 


4379.240 


80.612 






80.571 


680 


-0.109 


380 


-41.4 


4383.559 


81.388 


70 


81.458 


81.352 




-0.106 


382 


-40.5 


4399.780 


84.257 






84.228 


337 


-0.109 


390 


-42.5 


4404.763 


85.126 


86 


85.212 


85.105 




-0.107 


392 


-41.9 


4406.654 


85.453 






85.432 


539 


-0.107 


394 


-42.2 


4415.137 


86.913 


93 


87.006 


86.898 




-0.108 


397 


-42.9 


4425.446 


88.664 






88.662 


759 


-0.097 


402 


-39.0 


4428.551 


89.198 






89.194 


296 


-0.102 


404 


-41.2 


4430.624 


89.536 






89.535 


636 


-0.101 


404 


-40.8 


4434.969 


90.270 






90.270 


372 


-0.102 


406 


-41.4 


4435.690 


90.380 






90.378 


482 


-0.104 


406 


-42.2 


4442.351 


91.482 


108 


91.590 


91.491 




-0.099 


411 


-40.7 


4443.814 


91.724 






91.732 


831 


-0.099 


412 


-40.8 


4447.730 


92.365 






92.375 


473 


-0.098 


413 


-40.5 


4459.140 


94.216 


114 


94.330 


94.238 




-0.092 


417 


-38.4 


4476.023 


96.906 


127 


97.033 


96.940 




-0.093 


426 


-39.6 


4482.217 


97.872 


131 


98.003 


97.905 




-0.098 


428 


-41.9 


4494.575 


99.782 


138 


99.920 


99.820 




-0.100 


434 


-43.4 




- 35) 


1449.9 














-41 


.43 












Scs 


lie = +0 
va = +21 
vd = - 


.13 
.82 
.07 












Obse 


jrved V 


-19 


55 km 



144 THE BINARY STARS 

relation deduced on page 126, v the observed radial velocity is 
obtained by multiplying AX for each line by its corresponding 
factor V. = 299,796/X. 

Each spectrogram may be reduced in the manner outlined 
above, and some observers prefer to follow this method rigor- 
ously for each stellar spectrogram. When this is done the 
process is simplified by carrying through the computation in 
wave lengths, so that the displacement is expressed at once 
in Angstroms. 

Since for the same spectrograph the form of the dispersion 
„urve differs but slightly for different temperatures, a simple 
and practical method of reduction is offered by the following 
procedure: A standard dispersion curve is computed once for 
all, according to the method described above, from measures 
of a solar spectrogram. With the aid of this all other spectro- 
grams taken with the spectrograph may be quickly and easily 
reduced. It is convenient to put this standard curve in the 
form of a dispersion table in which are entered the normal wave 
lengths of the comparison and stellar lines used for stars of differ- 
ent spectral classes, and the micrometer readings corresponding 
to these wave lengths. In this standard table are given also the 
values of rV, for each line. Columns I, II, and VIII, in the 
example, are taken from such a table. 

It is now only necessary to reduce the readings of the standard 
table to the dispersion of the plate, by plotting the differences 
between the observed and table readings of the comparison 
lines (recorded in column III in the example). From this 
curve the difference to be applied to the table reading for each 
star line is read off. In the sixth column are given the new 
table readings (for zero velocity) after this difference has been 
applied. When there are comparison lines corresponding to 
star lines some observers follow rigorously the process outlined, 
while others (as in the present example) take the difference 
between the readings of the two as the displacements. The 
last three columns contain, respectively, the displacements 
(* minus Comp. or Sup'd), the factor rV S) and the products 
of these two values, which are the relative radial velocities of 
star and observer as supplied by the lines measured. The 
mean of the measures for 40 lines gives as the observed radial 
velocity —41.43 km/sec. It will be noticed that the dispersion 
of the star plate is about 0.3 per cent greater than that of the 



THE RADIAL VELOCITY OF A STAR 145 

standard table, and consequently the factor r (computed for the 
table) is too large, and the numerical value of this velocity must 
be reduced by this amount. This is allowed for, in the example, 
as scale correction. In practice, it is convenient to have several 
standard tables corresponding to the dispersion of the spectro- 
graph at different temperatures. The one whose dispersion is 
nearest that of the star plate is selected for use. Experience has 
shown that the results obtained by the very simple method 
just described are of the same accuracy as those derived by 
the longer process of computing a dispersion curve for each 

plate. 

If the spectrograph is not provided with a curved slit, it is 
necessary to introduce a correction for the curvature of the 
lines. This correction may be computed from Ditscheiner's 
formula* or determined empirically from lines on a spectrogram 
of the Sun, on the assumption that the curve of each line is a 
parabola. The better method is to eliminate the source of 
this correction by the use of a curved slit. 

The observed radial velocity of a star is made up of the star's 
velocity, V, with reference to the solar system, and the velocity 
of the observer in the solar system. The latter consists of three 
components, which arise from (1) the revolution of the Earth 
around the Sun; (2) the rotation of the Earth on its axis; (3) 
the revolution of the Earth around the center of mass of the 
Earth-Moon system. This last component never exceeds 
±0.014 km/sec and may be neglected. The correction for the 
annual and diurnal motions of the Earth are readily computed 
from the formulas given by Campbell in Frost-Scheiner's Astro- 
nomical Spectroscopy (pages 338-345). The values for these 
in the example are given respectively under v a and v d . Hence, 
the observed radial velocity of a 2 Centauri with reference to the 
Sun on 1911, February 27.883 (Greenwich Mean Time) was 
— 19.55 km/sec. 

Methods of reduction which depend upon dispersion for- 
mulas require an accurate knowledge of the wave lengths of 
the lines used in both the comparison and stellar spectra. Accu- 
rate values of the absolute wave lengths are not required but 
their relative values must be well determined. For example, 

* Uber die Krummung der Spectrallinien, Sitz. Ber. d. Math. Klasse d. k. 
Akad. Wien, Bd. LI, Abth. II, 1865; also Frost-Scheiner, Astronomical 
Spectroscopy, p. 15, 1894. 



146 THE BINARY STARS 

a relative error of +0.01 A in the wave length of any line would 
produce an error in the velocity for that line of nearly a kilometer. 
Interferometer measures of the wave lengths in the spectra 
of a number of elements are now available, but for the wave 
lengths of solar lines it is still necessary to use the determinations 
by Rowland. Fortunately, we now possess accurate wave 
lengths of the lines in the spectra of most of the elements, and the 
Revision of Rowland's Preliminary Table of Solar Wave Lengths 
has furnished a homogeneous set of data for the lines of the 
Sun's spectrum. A serious difficulty, however, arises for stellar 
lines, from the fact that stellar spectrographs have not sufficient 
resolution to separate lines which were measured as individual 
lines in the solar spectrum with the more powerful instruments 
employed by Rowland and his successors. It is the practice of 
many observers, where two lines merge to form one line in the 
star spectrum, to take the mean of the wave lengths of the 
component lines, weighted according to the intensities given by 
Rowland for those lines in the Sun. Wave lengths based on 
estimates of intensity should naturally be regarded with suspicion, 
and in fact we do not know, until the entire plate has been 
reduced, whether we have chosen an erroneous wave length or 
not. It is well known that various stellar lines and blends behave 
differently for stars of different types. The lines in solar type 
stars are assumed to have the same wave lengths as similar 
lines in the Sun. In the case of stars of other spectral classes, 
the solar lines which occur can be used in determining the 
wave lengths of the nonsolar lines and blends. In this manner 
special tables are constructed for stars of different types. 

When spectrographs of lower dispersion and resolution than 
those of three prisms are employed for the measurement of solar 
and later-type spectra, the effect of uncertainties in wave length 
of the stellar lines, due to blends, becomes very serious. The 
two methods of measurement and reduction which follow 
eliminate the sources of error incident to the use of blends,* and 
erroneous wave lengths as far as it is possible to do so. The 
first is that due to Prof. R. H. Curtiss and is called by him 
the velocity standard method. In principle it amounts to a 
determination of the wave lengths of the lines in the spectrum 
of a source whose radial velocity is known made with the particu- 
lar spectrograph which is to be used for measures of stellar spectra 
of this same class. Thus for the measures of spectra of the 



THE RADIAL VELOCITY OF A STAR 147 

solar type, a table similar to the one we have described is formed. 
The micrometer readings in this table, however, are not com- 
puted from assumed wave lengths, but are the mean of the actual 
settings, on comparison and solar lines, obtained on several 
spectrograms of the Sun. These standard plates are produced 
as nearly as possible under the same conditions as the stellar 
plates to be measured. The procedure in the reduction of the 
measures by means of this table is then the same as that described. 
It is necessary, of course, to correct the measured stellar velocity 
for the radial velocity of the source when the standard spectro- 
grams were taken. Standard tables for the reduction of measures 
of stars of other spectral classes may be formed in a similar 
manner, using as the standard sources stars whose radial veloci- 
ties are well determined. 

The second method is due to Prof. Hartmann and is in prin- 
ciple the same as the preceding one, except that the star plate 
is referred directly to the standard plate on a special measuring 
microscope, known as the spectrocomparator. The instru- 
ment is provided with two plate carriages, one of which is 
movable. On one of the carriages the star plate is placed 
and on the other, which is provided with a fine micrometer 
screw, is a standard plate of the Sun (taken with the stellar 
spectrograph). The microscope has two objectives so arranged 
that the images of portions of the two plates are brought, by 
means of total reflection prisms and a reflecting surface, to 
focus in the same plane and in the field of one eyepiece. By 
means of a silvered strip on the surface of one prism, the central 
portion of the Sun's spectrum is cut out and the star spectrum 
thrown into its place. In a similar manner, central strips of the 
comparison spectra of the Sun plate are replaced by those of 
the comparison spectra of the star plate. An ingenious arrange- 
ment of the microscopes permits of equalizing the scale of the 
two plates, by changing the relative magnifying powers of the 
two objectives. The method of measurement is, then, after 
proper alignment of the plates, to bring corresponding sections 
of the two plates into the field of the microscope, and by means 
of the micrometer screw set the corresponding lines of the 
comparison spectra in the same straight line. A setting is then 
made with the corresponding lines of the solar and star spectra 
in the same straight line. The difference between the micrometer 
readings in the two positions is the displacement of the star lines 



148 THE BINARY STARS 

relative to the solar lines. In practice it is found sufficient to 
divide the length of the spectrum into about 15 sections, for each 
of which these comparative settings are made. The mean of 
the displacements, obtained with the plates in the direct and 
reverse positions, when multiplied by the rV 8 for each section, 
gives for each the value V* — V , where V* is the radial velocity 
of the star and T that of the Sun. Theoretically, the values 
of V* — V Q should receive weights proportional to 1/rV, in 
taking the mean. Although this correction is negligible, except 
where an extent of spectrum of 400 or 500A is used, its introduc- 
tion leads to a very simple method of computation. Take the 
sum of the displacements in the direct and reverse measures 

and multiply by a factor / = — • The product is equal to 

the weighted mean of the values V* - V for each section. 
This, corrected for the velocity of the original Sun plate (Fo), 
gives the radial velocity of the star relative to the observer. 
The reduction to the Sun is made in the usual way. The 
factor / is a constant so long as the same regions are used, and its 
values may be computed for all combinations of the regions that 
are used. The great advantage of the method, aside from those 
which it possesses in common with the velocity standard method, 
is that we are able to measure and reduce in an hour a plate of a 
star rich in lines, and practically utilize all the material on the 
plate. With the older methods, to make such a comprehensive 
measure and reduction, i.e., to utilize all of the lines on the plate, 
would require one or two days. 

For the measures of spectra of a type other than the solar 
it is necessary to select for the standard plate a spectrogram 
of a star of that particular spectral class. In order to obtain 
the velocity for this standard spectrogram, it should be meas- 
ured and reduced, either by the method first described or perhaps 
preferably by the velocity standard method. The adopted 
value should be the mean of the measures made by several 
different observers. 

The spectrocomparator offers a very efficient method for 
determining the differences in velocities of the same star, by 
measuring a series of plates of the star with reference to one 
of these selected as a standard. 



THE RADIAL VELOCITY OF A STAR 149 

Five of the six elements of a spectroscopic binary orbit depend 
only upon the accurate determination of the relative radial veloci- 
ties given by the series of spectrograms. One of the most 
important applications of the Hartmann comparator is, therefore, 
to the measurement of plates of a spectroscopic binary. 

For the measure and reduction of spectrograms of stars of 
the earlier spectral classes, the use of the Cornu-Hartmann dis- 
persion formula will suffice, inasmuch as the spectra of such 
stars consist of lines due to the simple gases, the wave lengths 
of which have been accurately determined in the laboratory. 

The measure and reduction of spectrograms of stars of the 
solar and later classes of spectra are accomplished with great 
saving of time and labor, and by a method free from some of 
the uncertainties of wave lengths, by the use of the spectro- 
comparator. If the observer is not provided with such an 
instrument the velocity standard method is preferable to the 
use of the dispersion formulas, at least until a system of stellar 
wave lengths of the requisite accuracy is available. 

To the reader who has followed the long and intricate process 
of determining the radial velocity of a star, the question will 
naturally occur, how do we know that the final result repre- 
sents the star's velocity? Obviously, the final test of the 
method is its ability to reproduce known velocities. Fortu- 
nately, we have at hand a means of making such a test. Since 
the orbital elements of the inner planets of the solar system 
are well determined, we can readily compute the radial velocity 
of one of these with reference to the Earth at any given time. 
It is only necessary, then, to observe the relative radial velocity 
of the planet and the Earth and compare this with the computed 
value at the time of observation. At the Lick Observatory 
spectrograms of Venus and of Mars are secured at frequent 
intervals with the stellar spectrograph and measured by the 
observers in the regular course of measuring stellar plates. 
With the three-prism spectrograph, described above, the observed 
and computed velocities of these two planets generally agree 
to within ±0.5 km, or the unavoidable error of measure. When 
the spectrograms are measured by several observers, the effects 
of personal equation are to some extent eliminated in the mean, 
and an agreement within a few tenths of a kilometer is to be 
expected. A continual check is thus afforded on the adjustments 
of the spectrograph and the measurement of the spectrograms. 



150 THE BINARY STARS 

References 

General 

Campbell, W. W.: Stellar Motions, Yale University Press, 1913. 
Eberhard, G.: Sternspektrographie und Bestimmung von Radialge- 
schwindigkeiten, Handbuch der Astrophysik, I, chap. 4, 1933. 

Instruments and Design 

Campbell, W. W.: The Mills Spectrograph, Ap. Jour. 8, 123, 1898. 

Frost, E. B., The Bruce Spectrograph, Ap. Jour. 16, 1, 1902. 

Hartmann, J.: Remarks on the Construction and Adjustment of Spectro- 
graphs. Ap. Jour. 11, 400, 1900; 12, 31, 1900. 

Keeler, J. E. : Elementary Principles Governing the Efficiency of Spectro- 
graphs for Astronomical Purposes, Sidereal Messenger 10, 433, 1891. 

Newall, H. F. : On the General Design of Spectrographs to be Attached to 
Equatorials of Large Aperture, Considered Chiefly from the Point of 
View of Tremor-discs, Mon. Not. R.A.S. 65, 608, 1905. 

Plaskett, J. S. : Spectrograph of the Dominion Astrophysical Observatory, 
Publ. Dominion Astroph. Obs. 1, 81, 1920. 

Vogel, H. C: Description of the Spectrographs for the Great Refractor at 
Potsdam, Ap. Jour. 11, 393, 1900. 

Wright, W. H. : Description of the Instruments and Methods of the D. 0. 
Mills Expedition, Publ. Lick. Obs. 9, 25, 1905. 

Methods op Measurement and Reduction 

Campbell, W. W. : The Reduction of Spectroscopic Observations of Motions 
in the Line of Sight, Astronomy and Astrophysics, 11, 319, 1892. Also, 
Frost-Scheiner, Astronomical Spectroscopy, p. 338. 

Curtiss, R. H. : A Proposed Method for the Measurement and Reduction of 
Spectrograms for the Determination of the Radial Velocities of Celestial 
Objects, Lick Obs. Bull. 3, 19, 1904; Ap. Jour. 20, 149, 1904. 

Hartmann, J. F.: Uber die Ausmessung und Reduction der Photograph- 
ischen Aufnahmen von Sternspectren, A.N. 156, 81, 1901. 

: A Simple Interpolation Formula for the Prismatic Spectrum, Ap. 

Jour. 8, 218, 1898. 

: The Spectro-comparator. Ap. Jour. 24, 285, 1906; Publ. Astroph. 



Obs., Potsdam, 18, 5, 1908. 

[For a modern discussion of spectrum analysis see : J. A. Hynek, ed., Astro- 
physics, pp. 12-258, 1951; R. H. Garstang, Peculiar Stars, R. A. S. Occasional 
Notes 3, 21, Nov. 1959; and Helmut A. Abt, A Discussion of Spectral Classi- 
fication, Ap. Jour. Supplement Series 8, 75, Apr. 1963. — J.T.K.] 



CHAPTER VI 

THE ORBIT OF A SPECTROSCOPIC BINARY STAR 

The problem of determining the orbit of a binary system 
from measures of radial velocity, made in the manner described 
in the previous chapter, differs in several important particulars 
from that of computing an orbit from micrometric measures of 
position angle and distance. It has been shown that microm- 
eter measures provide the data from which the projection of 
the orbit of the companion star with respect to its primary can 
be drawn, the true relative orbit following, correct in propor- 
tions but of unknown linear dimensions. The radial velocities, 
on the other hand, when plotted against the times, produce a 
periodic curve, having the general appearance of a distorted 
sine curve; from this curve we are to find the elements of the 
true orbit of the star with respect to the center of gravity of 
the system of which it forms one component.* 

Figure 9 illustrates the conditions of the problem. Let the 
.XT-plane be tangent to the celestial sphere at the center of 
motion, and let the Z-axis, perpendicular to the XT-plane, be 
parallel to the line of .sight along which the radial velocities 
are measured. The velocities are considered positive (+) when 
the star is receding from, and negative ( — ) when it is approaching 
the observer. The orientation of the X- and F-axes remains 
unknown. Let PSA be the true orbit of the star with respect 
to the center of motion and let the orbit plane intersect the 
ZF-plane in the line NN'. 

Then, when the star is at any point S in its orbit, its distance 
z from the .XT-plane will be 

z = r sin i sin (v + «) 

* It is here assumed that the spectrum of only one component is visible; 
when both components give spectra, we may determine the relative orbit 
of one with respect to the other, using the same formulas but changing the 
value of the constant of attraction. The relative and absolute orbits are, 
of course, similar in every respect. 

151 



152 



THE BINARY STARS 



the symbols in the right-hand member of the equation having 
the same significance as in the case of a visual binary star. 

The spectrograph, however, does not give us the distances of 
the star from the .XY-plane, but the velocities of its approach 




Fig. 9. — The spectroscopic binary star problem. 

to, or recession from this plane, generally expressed in kilometers 
per second. The radial velocity at point S is equal to dz/dt, 
and is therefore expressed by 

dz . . . . . .dr . . . , . .dv 

-37 = sin 1 sin (y + «) -77 + r sin t cos (v + w)-j- 



From the known laws of motion in an ellipse we have 

dr _ fiae sin v 
dt 



dt 

and therefore 
dz 



dv _ m<*(1 + e cos v) 



vr= 



vr= 



na sin 1 



dt y/\ - 



[e cos co -f- cos (v + «)] 



(1) 



which is the fundamental equation connecting the radial velocities 

with the elements of the orbit. * 

* In place of (v + w) the symbol u ( = the argument of the latitude) is 

dr du 

often used, the expressions for -n and r-^7 written 



dt 



dt 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 153 

The observed velocities evidently contain the velocity, V, 
of the center of mass of the system, which is a constant quan- 
tity for any given simple binary system,* as well as the variable 
velocities due to the star's orbital motion and the quantity V 
must therefore be subtracted from the observed values to make 
them purely periodic. In other words, the velocity curve is 
purely periodic only with respect to a line representing the 
velocity of the system as a whole. This line is called the F-axis. 

Equation (1) applies only to the velocities counted from the 
,7-axis. If dt/dt represents the velocity as actually observed 
(i.e., the velocity referred to the zero-axis) we shall have the 
relation f 

f= F + ^ = F + J^±[ecos (a + cos (v + «)]. (la) 
dt dt "vl — e 2 

Methods of determining the position of the F-axis will be 
given later; for the present we shall assume it to be known. 

Five constants enter the right hand member of Eq. (1), viz., 
a sin i, e, n, co, and (through v) T. These express the five orbit 
elements which it is possible to determine by measures of radial 
velocity. 

Since the inclination of the orbit plane is not determinable, 
the value of a, the semimajor axis, must also remain unknown. 
It is therefore customary to regard the function a sin * as an 

I-^» to <»— >• and ^- / -7 5 = 7j 11+eeos( "-" ,) 

and hence the fundamental equation in the form 

dz I ■ • / i \ 

-j- = — -=. sin % (cos u + e cos w). 
dt ^Tp 

In these equations p [ = o(l - e 2 )] is the semi-parameter of the true 
ellipse and / denotes the constant of attraction, which, when the spectrum 
of only one component is visible, and the motion is determined with refer- 
ence to the center of mass of the system, takes the form kmfi/{m + mi), 
k being the Gaussian constant; when both spectra are visible and the motion 
of one star with respect to the other is determined, / = ky/m + mi- It is 
clear that the form of the fundamental equation will be the same whatever 
value we may assign to / and the constant of attraction may therefore be 
disregarded until the question of the relative masses in the system comes 
up for discussion. 

* In a triple or multiple system, this quantity will itself be variable, 
f The symbol y is often used for the velocity of the system instead of V. 



154 THE BINARY STARS 

element. Further, it is clear that the position of the line of 
nodes cannot be determined, though we can find the times when 
the star passes through each of the nodal points. The various 
elements have the same definitions as in the case of visual 
binary star orbits (see page 78), except that the angle « in spec- 
troscopic binary orbits is always measured from the ascending 
node, the node at which the star is moving away from the observer. 
It will be seen later that the radial velocity has its maximum 
positive value at this node and its minimum positive value (or 
maximum negative value) at the descending node. It should 
also be noted that the unit of time for fi (and therefore for P) 
is the day, not the year as in visual binary orbits. 

Theoretically, values of the radial velocity at five different 
times suffice for the complete solution of Eq. (1); practically, 
no computer undertakes an orbit until a considerable number of 
measures is available which give the velocities at short intervals 
throughout the entire revolution period. To secure a satis- 
factory distribution of the observations a preliminary value 
of the period is necessary and such a value can ordinarily be 
obtained without difficulty by plotting the early observations 
on coordinate paper, taking the times, expressed in Julian 
days and decimals of a day, as abscissas and the velocities, 
expressed in kilometers per second, as ordinates. A convenient 
epoch as origin for the period is selected near the beginning of 
the series, preferably one corresponding to a point of maximum 
or minimum velocity. If later measures indicate that the 
period is in error, a new period which is a submultiple of the 
original one will often prove satisfactory. In difficult cases, 
the following artifice may be found helpful.* Copy on trans- 
parent paper from one-third to one-half of the series of observed 
points, choosing the time interval best covered by observa- 
tion ; slide the copy along the original plot, keeping the time axis 
in coincidence, until some point on the copy falls approximately 
upon a different point in the original at which the velocity is 
changing in the same direction. The time interval between the 

* This was suggested to me by Dr. R. K. Young who said that it had been 
used with good results by several computers of binary star orbits at the 
Dominion Observatory. No prior mention of the device has been found in 
print and its author is unknown to me. Its usefulness arises from the fact 
that, in effect, it doubles the number of observations for a given time 
interval. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 155 

two points is evidently equal to the period or a multiple of the 
period. 

Schlesinger* has published a criterion that may be applied 
to advantage in cases where an observer has accumulated 
many plates of a star which apparently shows variable radial 
velocity without being able to determine any period. It con- 
sists in constructing a frequency curve for the velocities by 
"dividing the total range exhibited by the measured velocities 
into successive groups of equal extent, say 3 km each, and 
then counting the number of velocities that fall within these 
groups. Regarding these numbers as ordinates, we plot them 
and join the ends by a smooth curve." This curve is com- 
pared with the well-known error curve; if the two are the same, 
within reasonable limits, we may conclude that the differences 
in the measured velocities are due to errors of observation, 
and afford no support for the assumption that the star is a 
spectroscopic binary. If the two curves differ, the star is a 
binary and the form of the frequency curve will give an idea 
as to the general character of the orbit and frequently furnish a 
clew to the period. For Schlesinger shows that circular orbits, 
elliptic orbits with periastron at descending node, elliptic orbits 
with periastron at ascending node, and elliptic orbits with 
periastron removed 90° from the nodes, all have characteristic 
frequency curves which differ in form from the error curve. 
When the nature of the frequency curve has shown to which of 
these classes the orbit in question belongs, it becomes very 
much easier to decide upon the epochs for the various observed 
velocities, and thus upon an approximate value for the period. 

When the period is approximately known all of the observa- 
tions may be reduced to a single revolution by subtracting 

* Ap. Jour. 41, 162, 1915. In his paper on the "Orbit of the Spectro- 
scopic Binary x Aurigae" (Jour. R.A.S.C. 10, 358, 1916), Young shows that 
the errors of measurement may affect the expected distribution in such a 
manner as to mask to a considerable degree the presence of the orbital 
variation. If possible, all of the spectrograms used in the determination 
of a particular orbit should be secured with the same instrument and meas- 
ured by the same person, to avoid the effect of systematic errors of observa- 
tion and of the personal equation in measurement. The small systematic 
differences between the radial velocities of stars made at different observa- 
tories have been discussed by several investigators. See, for example, Dr. 
J. H. Moore's Introduction to his General Catalogue of the Radial Velocities 
of Stars, Nebulae, and Clusters (Publ. Lick Obs. 18, 1932). 



156 THE BINARY STARS 

multiples of the period from the later dates. A preliminary 
curve is drawn to represent the plotted positions as closely as 
possible. The deviations from the curve at points near the 
mean of the maximum and minimum velocities, where a change 
in the periodic time will have the greatest effect, will indicate 
advisable changes in the assumed period and these are readily 
found by dividing the deviations of such critical observations, 
expressed in time, by the number of revolutions elapsed. A 
second curve is then drawn whose periodic time will generally 
be very close to the true value. In practice it will frequently 
happen that two or three measures of the radial velocity of a 
star are available which were made (perhaps at another obser- 
vatory) several years before the series of spectrograms for the 
orbit computation is begun. When a fairly good value of 
the period has been found from the later series, these early plates 
will determine its true value with high precision. Generally 
they are not used in finding the other orbit elements. 

When the period has been determined as accurately as possible 
and a series of spectrograms has been accumulated giving 
the velocities at points well distributed throughout the entire 
period, the most probable curve is drawn, by estimation, through 
the points as plotted, and, if the ingenious methods of super- 
position devised by Schwarzschild and Zurhellen are to be 
used, the curve should be prolonged through a revolution and 
a half. The plotted points used for this curve should repre- 
sent normal positions, formed by combining several velocities 
observed at very nearly the same orbit phase, whenever the 
number of observations is sufficient to permit such combina- 
tions. In making the combinations, the question of weights 
arises, and here the practice of computers varies considerably, 
for several factors enter. The character of the lines on the 
spectrograms, broad or narrow, sharp or ill-defined, strong or 
weak, is one factor; the number of lines is another; if the plates 
have been taken with different telescopes and spectrographs, 
a third factor is introduced. These must all be considered 
in assigning the weights to each plate. The only direction 
that can be given is the general one to use a rather simple system 
of weighting. It will rarely be of advantage to assign fractional 
weights, or to use a range of weights greater than, say, four units. 
The weights should, of course, be assigned to each plate, at the 
time of measurement. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 157 

The errors in drawing the most probable curve have con- 
siderable effect upon the accuracy of the determination of the 
elements. At best the curve is not likely to be a perfect repre- 
sentation of the elliptic motion which caused it since it is natural 
to bend the curve slightly in or out at different points to satisfy 
the more or less exact observations. This difficulty is inherent 
and for it there is apparently no remedy other than that of 
testing the first orbit by a trial ephemeris and making the small 
changes in the elements which are indicated by the residuals.* 



OKm.- 




Zero 
axis 



10 2030405060 70 80 90 100 U0120c*gu 
6476.0 U U 

Fio. 10. — Velocity curve of k Velorum. 

If Fig. 10 represents a velocity curve, it is evident from 
Eq. (1) that the points A and B correspond respectively to 
the ascending and descending nodes of the star's orbit, for 
at the times of nodal passage we have (v + «) = 0°, and 
(v + a>) = 180°, respectively, and therefore cos (v + «) = ±1. 
The radial velocity thus reaches its maximum and minimum 
values at the nodal points. 

Taking A and B as the magnitudes of the curve ordinates 
at the points of maximum and minimum reckoned from the 
7-axis, regarding B a s a po sitive quantity and writing for 
brevity K = na sin ijy/\ — e 2 , we have 

A = K(l + e cos «) 
B = K(l — e cos w) 

* King's method affords a graphical test of the first orbit found, see page 
170. 



158 THE BINARY STARS 

and therefore 



A +B 

2 
A- B 

2 
A ~ B 



= K 

— Ke cos co } (2) 

= e cos w 



A + B 
Hence we may write Eq. (1) in the form 

■£ = K[e cos co + cos (v + co)] = — 1 ±— cos (v + co) (3) 

and Eq. (la) in the form 

^- = K H ^ 1 2 — C0S ( w + w ) = ^i + 

— 2 — cos (t; + co). (3a) 

if is therefore the half amplitude of the velocity curve. 

Up to the point now reached practically all methods of spec- 
troscopic orbit determination are identical. But when the 
fundamental relations are given as above, and the curve has 
been drawn, various methods are available for computing the 
orbit elements, other than the period, which is assumed to be 
known. 

Of these, the method devised by Lehmann-FilhSs will first 
be presented, essentially in full; other methods will then be 
treated in less detail. The student who desires to study the 
various methods more fully is referred to the important papers 
given in the references at the end of the chapter. 

METHOD OF LEHMANN-FILHES 

Given the observations, and the velocity curve drawn with 
the value of P assumed as known, the first step is to fix the 
F-axis, the line defining the velocity of the center of gravity 
of the system. This is found by the condition that the integral 
of dz/dt, that is, the area of the velocity curve, must be equal 
for the portions of the curve above and below the 7-axis. By 
far the easiest method of performing this integration is to use 
a planimeter. A line, approximately correct, is first drawn by 
estimation; the areas contained between it and the curve above 
and below are measured, and the difference between the two 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 159 

is taken. The position of the axis is then shifted to eliminate 
this difference, and the measures are repeated. It will rarely 
be necessary to make more than one correction to secure an 
accurate value of the position of the F-axis, which, by this 
method, depends upon the entire curve. 

If a planimeter is not available, the areas above and below 
the axis may be equalized by using coordinate paper for the 
plot of the curve and counting the small squares in each area. 
Approximate mechanical integration, as advised by Lehmann- 
Filh6s, may also be resorted to, by those who enjoy this form 
of recreation. 

Having found the F-axis, the ordinates to it are next drawn 
from the points of maximum and minimum velocity, A and B. 
It is at this point, as Curtis says, that the method is weakest, 
for slight errors in fixing the position of A and B may easily 
arise. It is well to apply the check afforded by the requirement 
that area AaC (Fig. 10) must equal CbB and DaA equal BID. 
Since C and D lie on the 7-axis the velocities at these points 
are zero, hence from Eqs. (3) and (2) we have for dz/dt at these 
points 

cos (v + w) = - A , B = ~e cos co (4) 

If Vi is the true anomaly corresponding to the point C, which 

is traversed by the star on the way from the ascending to the 

descending node, and « 2 , the true anomaly for the point D, 

sin (vi + «) will be positive, sin (v 2 + «) negative, and we 

shall have 

A - B , . x A - B ) 

cos (t>i + w ) = - A + B > cos (v* + »; - - A + 5 | 

sm (»! + «) = A+B > sm («* + «) = ~ A + b 

Let Zi and Z 2 denote the areas* AaC and bBD (Fig. 10), respec- 
tively, and let n and r 2 be the radii vectores for the points C 
and D. 
Then 

Z x = r x sin i sin (v x + o>) 

Z 2 = r 2 sin i sin (i> 2 -f <a) = -r 2 sin i sin (vi + co) 

* These areas represent the distances of the star from the XF-plane at the 
points in its orbit corresponding to (»i + «) and (» 2 + »)• 



(5) 



160 THE BINARY STARS 

and therefore 

— — \ = - 1 = 1 + g COS V 2 

Z 2 r 2 1 + e cos Vi ^ ' 

since r = [a(l - e 2 )]/(l + e cos 0). Write (v + w - &>) for v, 
in Eq. (6), expand, and reduce, with the aid of the relations in 
Eqs. (5) and (4), and we have 

_Zi sin (vi + w) — e sin a> 
Z 2 sin (t>i + w) + e sin w 
whence 

Equation (7) and the last of Eq. (2) determine e and o>. The 
values of A and B are taken from the curve, and the areas Z\ 
and Z 2 are quickly integrated from the curve portions AaC 
and bBD by means of a planimeter, the latter area being regarded 
as negative in sign. Since the areas enter as a ratio, the unit 
of area used is entirely immaterial. 

At the time of periastron passage v — 0°; hence from Eq. (3) 
we have 

£ = KQ. + e) cos a, (8) 

which gives the ordinate corresponding to the point of peri- 
astron passage. Two points of the curve will have the same 
ordinate, but since (v + o>) equals 0°, 180°, and 360° for the 
points A, B, and Ai, respectively, there will be no ambiguity 
as to the position of the periastron point. The abscissa of 
this point, properly combined with the epoch chosen for the 
beginning of the curve, defines T, the time of periastron pas- 
sage. Instead of using Eq. (8) we may find T by determining E 
for the point C for which the value of v is known, and then 
employ the formulas 



tan }^E = -J e tan }4v) 

V +e C (8a) 



T = i- E ~ e sin E 



or, if the eccentricity is less than 0.77, the value of M correspond- 
ing to v may be taken directly from the Allegheny Tables, and T 
found from the relation 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 161 

M = n(t - T). (86) 

Such procedure is especially advisable when the periastron 
points fall near point A or B on the curve. 
By definition (page 157) we have 

fj.a sin i 
K. — 



and hence 

A_+ B V 1 - e 2 
a sin i 2 a sin i 

from which we may find the value of the product a sin i. Since 
the unit of time for A and B is the second, while for m it is the 
day, the factor 86,400 must be introduced. Our equation 
then becomes 

a sin i = 86,400 - \/T^7 2 = (4.13833)KP y/\ - « 2 (9) 

the number in parentheses being the logarithm of the quotient 
86400 -s- 2tt. 

Summarizing, the practical procedure is: 

1. Find the period as accurately as possible by successive 
trials and plot the most probable velocity curve on the basis of 
normal places. 

2. Find the position of the 7-axis by integration of areas, using 
the planimeter, if available. 

3. Measure the ordinates for points A and B and find the areas 
of AaC and bBD expressed in any convenient units. 

4. From Eqs. (2) and (7) determine K, e, and <i. 

5. From Eq. (8), or by calculation from the value of v, for 
the point C, determine T. 

6. From Eq. (9) determine a sin i. 

To test the elements by comparison with the observations, 
we compute the radial velocity for each date by the formulas: 



M = fx(t - T) = E - e sin E 
tan Y 2 v = Jr— tan % E 

$ = V + Ke cos to + K cos (v + «) 
at 



(10) 



162 



THE BINARY STARS 



The value of v for each value of M may be taken directly from 
the Allegheny Tables, if e is less than 0.77. 

To illustrate Lehmann-FilheV method I have chosen the orbit 
computed for k Velorum, by H. D. Curtis, the velocity curve 
for which is given in Fig. 10. 

The observations used were as follows: 



Julian Day, G. M. T. 


Velocity 


Julian Day, G. M. T. 


Velocity 


2,416,546.739 


+68. 5^ 


_ _ 

2,417,686.591 


+33. 8*™ 


60.703 


+12.9 


91.572 


+38.2 


97.651 


+65.7 


92.545 


+43.2 


6,912.601 


+53.3 


96.480 


+46.7 


7,587.844 


+58.6 


7,701.494 


+52.7 


88.788 


+57.9 


41.466 


+22.1 


90.829 


+58.5 


46.463 


+ 0.3 


91.824 


+64.8 


49.470 


- 7.6 


97.788 


+65.8 


50.479 


- 8.8 


7,609.790 


+62.0 


51.463 


-13.3 


54.534 


-21.0 


53.457 


-19.2 


55.556 


-19.2 


58.451 


-29.0 


58.570 


-15.2 


59.460 


-24.6 


59.545 


-14.5 







The small circles representing the first four observations, 
which are important in determining the period, owing to their 
distance in time from the later ones, are barred in the diagram. 
The period, P, was assumed to be 116.65 days, and the begin- 
ning of the curve is at Julian day 2,416,476.0. This is not 
exactly at a minimum, as may be seen from the diagram. 

From measures of the curve we find 



A = 46.3 

B = 46.9 

A + B = 93.2 

A - B = -0.6 

K = (A + B)/2 = 46.6 



Zi = AaC = +0.168 
Z 2 = bBD = -0.259 
Z 2 + Zx = -0.091 
Z 2 - Zt 0.427 



the figures for area being expressed in decimals of the unit of 
area for the planimeter employed. 
The solution of Eqs. (2), (7), and (8) then proceeds as follows: 

0.3010 
1.6684 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 163 

colog (A + B) 8.0306 

log (Z 2 + Zi) 8.9560n 

colog (Zi - Zi) 0.3696n 

log e sin co 9.3286 

log (A - B) 9.7782n 



log ^ ~ B b =\ cos « 7.8088n 



log tan co 1.5198n 

co 91?73 
log sin co 9.9998 
log e 9.3288 
e 0.21 
log (1 + e) 0.0828 
log cos co 8.4800n 
log K 1.6684 

log cf °- 2312n 
ordinate p — 1.7 km 
.*. from curve t p = 98.4 

T* = J. D. 2,416,457.75 
log const. 4.1383 
log it 1.6684 
logP 2.0669 

logVT^e" 9 " 02 
log a sin i 7.8638 

a sin i 73,000,000+ km 

The preliminary values thus obtained are next tested by 
comparing the velocities derived from them by Eqs. (10) with 
the observed velocities. To illustrate, let us compute the 
velocity for J. D. 2,416,496.0, twenty days after the origin 
adopted in our curve. We have 

t = 2,416,496.0 log cos (v + co) 9.8277n 

t - T = +38.0 log K 1.6684 

log (t-T) 1-57978 1.4961n 

log n 0.48942 g cos (v + co) - 31.3 km 

* T is here taken one revolution earlier than the date for the periastron 
point marked on the curve. Using Eq. (8a) or (86) we obtain T = J. D. 
2,416,458.0 which is adopted. 



164 THE BINARY STARS 

M 117?27 4-Zj _ .3 

v 136.01 V + 20.7 

v + a, 227.74 * = -10.9 km 

In this manner we compute as many velocities as necessary to 
obtain a curve for comparison with the observed velocity curve. 
In the present instance this was done for every tenth day, and 
the results plotted as heavy black dots in the figure. By noting 
the discrepancies, it appears that the branch on the apastron 
side of the computed curve, if drawn, would be a little too 
sloping, the other branch too steep, which indicates that the 
computed value of e is a little too large. Changing this ele- 
ment and making the corresponding slight changes required 
in T and o>, the test was repeated and after a few trials the 
following elements were adopted as best representing the 
observations : 

V = +21.9 km 
P = 116.65 days 
e = 0.19 
K = 46.5 
w = 96?23 

T = J. D. 2,416,459.0 
a sin i = 73,000,000 km 

The correction to the value of V was found last of all from 
the residuals of the final ephemeris by the simple formula 
[v]/n, where n is the number of observations and v the residual, 
(o — c). The residuals from the final ephemeris and the final 
curve may be found in Lick Observatory Bulletin, No. 122, 1907. 

Lehmann-FilheV method may be termed the classical one, 
and it is probably more generally used than any other. The 
method proposed by Rambaut is considerably longer and 
more involved than the later ones, and for that reason will not 
be described here. Wilsing's method, as originally published, 
was suitable only for orbits of small eccentricity, but Russell 
later extended it to make it applicable to larger eccentricities 
as well. This method is purely analytical, consisting in finding 
a Fourier's series for the velocity in terms of the elements. It 
should be very useful in special cases, particularly when the 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 165 

period is so nearly a year that one part of the velocity curve 
is not represented by any observations; but it is considerably 
longer, in time consumed, than the method of Lehmann-FillwSs 
and other geometrical methods to be described presently and 
will not be further considered here. 

Certain features of the methods proposed by Schwarzschild 
and Zurhellen are both ingenious and practical. The following 
account of them is taken in substance from Curtis' article 
already referred to. 

SCHWARZSCHUD'S METHOD 

Given the velocity curve and the period, Schwarzschild first 
determines the time of periastron passage. Let Mi and M 2 be 
the observed velocities (i.e., the velocities measured from the 
zero-axis) of maximum and minimum, and draw the line whose 
ordinate is (Mi + M 2 )/2. This line is the mean axis. Mark 
upon it the points corresponding to P/2 and 3P/2; then lay a 
piece of semitransparent paper over the plot, copy upon it the 
curve together with the mean axis and mark also the points 0, 
P/2, P, and 3P/2. Shift the copy bodily along the mean axis for 
the 'distance P/2, and then rotate it 180° about this axis, i.e., 
turn the copy face downward on the original curve keeping 
the mean axis in coincidence and bring the point or P of the 
copy over the point P/2 of the original. The curves will then 
cut each other in at least four points, and, in general, in four 
points only. These will fall into two pairs, the points of each 
pair separated by an abscissa interval P/2. The points of one 
pair will be on different branches of the velocity curve, and it 
is easy to see that, if v x and v 2 represent their true anomalies, 
we shall have v 2 = v x + 180°. Now the only two points in the 
true orbit which are separated by one-half a revolution and for 
which at the same time this relation of the true anomalies 
holds are the points of periastron and apastron passage. Hence, 
to select these points, choose the two points of intersection of 
the curve and its copy which are separated by half a revolution 
and which he on different branches of the curve. To distinguish 
periastron from apastron we have the criteria: (1) at periastron 
the velocity curve is steeper with respect to the axis than at 
apastron; (2) the curve is for a shorter time on that side of the 
mean axis on which the point of periastron lies. 



166 THE BINARY STARS 

This method is exceedingly good except when the eccen- 
tricity is small. In this case, co and T are quite indeterminate 
and small errors in drawing the velocity curve will be very 
troublesome. The method of Lehmann-Filh6s is then to be 
preferred. 

Having the value of T, the value of co is next found as follows: 
From Eqs. (la) and (3) it is readily seen that the position of the 
mean axis is 

2 = V -\- Ke cos co = Vi 

and that, accordingly, the ordinate z' of any point measured 
from the mean axis is 

z' =j t - Vx = K cosO + «) (11) 

Now at periastron v = 0°, at apastron v = 180°. Hence, if 
we call the ordinates from the mean axis for these points z p ' 
and zj, we shall have 

cos « = jjjr' or cos a = **' 2K* < 12 ) 

from which to determine w. This method is at its best when 
co is near 90°. 

Zurhellen has simplified Schwarzschild's method of finding 
e, and we shall give this simpler form in connection with Zur- 
hellen's simple method of finding w. 

ZURHELLEN'S METHODS 

Zurhellen's method of determining co depends upon the rela- 
tions between the velocities for the two orbit points whose 
true anomalies are ±90°. From Eq. (11) we have, when 
v = -90°, 

Zi = +K sin co 

and when v = +90°, 

Zi = —K sin co 

Moreover, for these two points we have 

Ei = — E2 
M x = -M a 

(*i- T) = -(*,- T) 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 167 

Hence the two points are symmetrically placed with respect 
to the mean axis in the F-coordinate and with respect to the 
point of periastron passage in the X-coordinate. They may 
therefore be determined by rotating the curve copy through 
180° about the intersection of the ordinate of periastron with 
the mean axis, and noting the two points of intersection of the 
copy with the original curve. If the curve is prolonged through 
one and 'one-half revolutions, another point 180° from one of 
these, say at +270°, can be determined in similar manner 
and the location of all three can then be checked by drawing 
the lines connecting the point v = +270° with v = —90°, 
and v = —90° with v = +90°. These lines should cut the 
mean axis at its intersections with the ordinates of periastron 
and apastron, respectively. From the ordinates of the two 
points v = ± 90°, measured from the mean axis, we have 

sin co = — ,.„ ) or tan co = (12a) 

2K z P — z a 

from which to find co. The method is at its best when co is small. 
Zurhellen's simplification of Schwarzschild's method of find- 
ing e is also based upon the relations between the two points 
v = ±90°. Since 

tan Y 2 E = tan x Av tan (45° - l A4>) 

where <f> is the eccentric angle, we have, when v = + 90°, 

E x = -(90° - <6), E 2 = +(90° - c6) 

Similarly, 

sin c6 sin (90° - <f>) 



M x = -(90°- c6) + 



sin 1' 



M, = +(90° - ♦) - ***<*£&-*) 
and therefore 

*,_*,. w (h _ (i) . (180 . _ 2 „ _ smi^-2*) (13) 

The value of (fa — ii) may be read off directly from the dia- 
gram, and the value of (90° — 4>) can then be taken from the 
table for Eq. (13), computed by Schwarzschild, which is given 



168 



THE BINARY STARS 



below. Like the above method for finding o> this method is 
best when co is small. 



Schwarzschild's Table for the Equation 

360° 

2i\ — sin 2?7 = — — (< 2 — h) 





ti — ti 




ti — ti 




<2 — ti. 


V 


P 


V 


P 


V 


P 


0° 


0.0000 


30° 


0.0290 


60° 


0.1956 


1 


0.0000 


31 


0.0318 


61 


0.2040 


2 


0.0000 


32 


0.0348 


62 


0.2125 


3 


0.0000 


33 


0.0380 


63 


0.2213 


4 


0.0001 


34 


0.0414 


64 


0.2303 


5 


0.0001 


35 


0.0450 


65 


0.2393 


6 


0.0002 


36 


0.0488 


66 


0.2485 


7 


0.0004 


37 


0.0527 


67 


0.2578 


8 


0.0006 


38 


0.0568 


68 


0.2673 


9 


0.0008 


39 


0.0611 


69 


0.2769 


10 


0.0011 


40 


0.0656 


70 


0.2867 


11 


0.0015 


41 


0.0703 


71 


0.2966 


12 


0.0020 


42 


0.0751 


72 


0.3065 


13 


0.0025 


43 


0.0802 


73 


0.3166 


14 


0.0031 


44 


0.0855 


74 


0.3268 


15 


0.0038 


45 


0.0910 


75 


0.3371 


16 


0.0046 


46 


0.0967 


76 


0.3475 


17 


0.0055 


47 


0.1025 


77 


0.3581 


18 


0.0065 


48 


0.1085 


78 


0.3687 


19 


0.0077 


49 


0.1147 


79 


0.3793 


20 


0.0089 


50 


0.1212 


80 


0.3900 


21 


0.0103 


51 


0.1278 


81 


0.4008 


22 


0.0117 


52 


0.1346 


82 


0.4117 


23 


0.0133 


53 


0.1416 


83 


0.4226 


24 


0.0151 


54 


0.1488 


84 


0.4335 


25 


0.0170 


55 


0.1561 


85 


0.4446 


26 


0.0191 


56 


0.1636 


86 


0.4557 


27 


0.0213 


57 


0.1713 


87 


0.4667 


28 


0.0237 


58 


0.1792 


88 


0.4778 


29 


0.0262 


59 


0.1873 


89 


0.4889 


30 


0.0290 


60 


0.1956 


90 


0.5000 



Zurhellen also gives a method for finding the eccentricity 
by drawing the tangents to the curve at the points of perias- 
tron. These can be drawn quite accurately except when the 
periastron falls near a maximum or a minimum of the curve. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 169 

Slight changes in its position will then introduce considerable 
changes in the inclinations of the tangent lines. 

The expression for the slope of a tangent may be written 

dx _2ir dx _ 2t 1 dx 

dt ~ P "dM ~ P" ' 1 - e cos E ' dE 

where x (= dz/dt) represents the ordinate drawn to the F-axis. 
Also, by introducing the known values 

cos E — sin <f> . cos <f> sin E 

cos v = -= j=r> sin v = -2 = 

1 — e cos E 1 — e cos E 

and transforming and simplifying we may write the fundamental 
Eq. (3) in the form 

dz rr . cos <f> cos w cos E — sin co sin E 

x = -r = K cos <t> • = ™ 

dt 1 — e cos E 

Differentiating with respect to E, substituting and reducing, 
we have 

dx 27r„ , — cos <$> cos co sin E — sin co cos E + e sin co /1/IX 

j7 = -5- -K cos <£ r- p^ (14) 

dt P (1 — e cos Ey 

At periastron E = 0° and at apastron E = 180°, whence we 
have 



dx — 


2tK cos c/> sin 


CO 


dx 


-\-2irK cos 


4> sin co 


dtp 


P(l - e) 2 


— > 


dt a 


P(l + 


e) 2 


and therefore 














dx/dtp _ 
dx/dta 


(l + <0 2 

(1 - eY 


= "? 2 




whence 


f 


, - 9 


- 1 







•-,+1 (15) 

KING'S METHOD 

The methods of orbit computation so far described in this 
chapter all rest upon the curve drawn to represent as closely 
as possible the observed velocities and, at the same time, to 
satisfy the conditions for elliptic motion. Unless the measures 
are very precise, the first approximation will ordinarily not be 



170 THE BINARY STARS 

satisfactory. As stated on page 157, the only remedy is to 
compute an ephemeris from the elements and, on the basis of 
the residuals thus found, to draw a new curve. This process 
is sometimes repeated three or four times before a curve is 
found that will yield elements upon which a least squares 
solution may be based. 

The method devised by Dr. King, which is now to be pre- 
sented, aims to substitute a rapid graphical process for testing 





















































/' 


1\ 






^\ 






I 


; S 


^ 




k! 




\ 






1 v< 


A 




Y 














\ 
















\ 






/ 




\ 

























































Fig. 11. — King's orbit method. Graph for e = 0.75, « = 60°. 

the preliminary curve. Dr. King shows that a circle having 
its center on the mean axis and a radius equal to K, the semi- 
amplitude of the velocity curve, "may be used as the equivalent 
of the hodograph of observed velocities."* 

Let the velocity curve and the circle be drawn (see Fig. 11) 
and the abscissa distance corresponding to one revolution (P 
being assumed to be known) be divided into any convenient 
number of parts, say forty, f Now mark consecutive points 
on the circumference of the circle by drawing lines parallel to 

* For the proof of this relation the reader is referred to the original 
article in Ap. Jour. 27, 125, 1908. 

t An even number should be chosen, and it is obviously most convenient 
to make the drawing upon coordinate paper. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 171 

the mean axis at the intersections of the velocity curve with 
the ordinates corresponding to successive values of the abscissa 
and extending them to the circle. The circumference will 
be divided into forty unequal parts, but these inequalities will 
be found to vary uniformly. "The points will be close together 
in the vicinity of one point of the circle, and will gradually 
separate as we proceed in either direction therefrom, until 
at the diametrically opposite point they reach their maximum 
distance apart." These unequal arcs of the circle correspond 
to the increase in the true anomalies in the orbit in the equal 
time intervals, and therefore the point of widest separation 
of the circle divisions corresponds to periastron, that of least 
separation, to apastron. Further, the angle between the 
F-axis and the periastron-half of the diameter between these 
two points is equal to w. To locate the point of periastron 
on the velocity curve, find the intersection of the steeper branch 
of the curve with a line drawn from the periastron point on the 
circle parallel to the mean axis. 

It is evident that the division points of the circumference 
will be symmetrically disposed with respect to the apsidal 
diameter (the diameter joining periastron and apastron points) 
only when one of the division points in the line of abscissae 
corresponds to an apse. In general, the periastron point will 
lie within the longest division of the circumference, the apas- 
tron point within the shortest. If desired, the approximate 
position of one of these points may be used as a new origin 
from which to set off the fortieths of the period along the axis 
of abscissae, and two division points on the circle may then be 
brought into closer coincidence with the apsidal points. 

Since dv/dt varies inversely as the square of the distance from 
the focus, by measuring the lengths di and d 2 of the arcs at 
points where v equals Vi and v%, we have 

di _ (1 + e cos t>i) 2 , 
d 2 — (1 + e cos vz) 2 

and hence if the arcs are measured at the points of periastron 
and apastron where v equals 0° and 180°, respectively, 

\/di — \/^2 



d t \l - e) ' 



or e — 



■\Zd~i + y/di 
which determines e. 



(16) 



172 THE BINARY STARS 

It is generally sufficiently accurate to measure the chords 
instead of the arcs; when the eccentricity is high and the arcs 
at periastron are inconveniently long, additional points of division 
may readily be inserted. 

It will be observed that this process furnishes a more thorough test 
of the accuracy of the graph (velocity curve) than the method of equality 
of areas. If it is imperfect, the points on the circumference of the circle 
will not be distributed according to the regular order of increase or 
decrease of the included arcs. If an ordinate of the graph is too long or 
too short, the corresponding point on the circumference will be too near 
or too far from the vertical diameter. If the points of maximum and 
minimum velocity have not been well determined, the diameter of the 
circle will be too long or too short. In the former case all the points 
of the circumference will be crowded away from the vertical diameter; in 
the latter, toward it.* 

To test a given set of elements by comparison with the observa- 
tions proceed as follows: 

Construct a circular protractor on some semitransparent 
material (e.g., celluloid or linen tracing cloth) and divide it 
into forty parts by radii to points on the circumference repre- 
senting the true anomalies for the given value of e correspond- 
ing to every 9° of mean anomaly (i.e., to fortieths of the period). 
If the eccentricity is less than 0.77 the values of the true anomaly 
can be taken directly from the Allegheny Tables. 

On the plot of the orbit draw a circle of radius K with its 
center on the mean axis and draw its vertical diameter. Set 
the protractor upon the circle, making the centers coincide, 
and turn the apsidal diameter of the protractor until it makes 
an angle equal to w with the vertical diameter. Now note the 
points where the radial lines representing the anomalies inter- 
sect the circumference of the circle. The abscissa axis of the 
plot also having been divided into forty equal parts, erect 
perpendiculars at the points of division equal to the corre- 
sponding ordinates of the circle. A free-hand curve through the 
extremities of these perpendiculars (i.e., ordinates to the mean 
axis) gives the computed curve or ephemeris, and the residuals 
can be read directly from the plot. The advantage of using 
coordinate paper will be obvious. 

From the account just given it will appear that King's method 
is longer, or at least not shorter, than the others described 

* King, he. cit. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 173 

if only a single orbit is to be computed. But when orbit com- 
putation is to be taken up as a part of a regular program of 
work, the method has very decided advantages. It is then 
to be used as follows: 

Let a set of protractors be constructed on transparent cellu- 
loid with radii representing the divisions in true anomaly for 
every 9° of mean anomaly for the values e = 0.00, e = 0.05 
to e = 0.95. 

With the aid of these protractors draw curves on tracing 
linen representing orbits with all values of e from 0.00 to 0.95 
and all values of « from 0° to 360°. The intervals for e should 
be 0.05, save for the larger values which are seldom used, and 
for o), 15°. Practically, values of w to 90° will suffice, the curves 
for the values in the remaining quadrants being obtained by 
inverting the sheet and by looking through the linen from the 
back in the two positions. Given the protractors, a complete 
set of curves may be constructed in about 10 hours' time. 

Having such a set of curves, plot the normal places for any 
given binary star on the same scale as these curves in time and 
in velocity.* Now place the standard curves upon the plot 
until one is found that fits the observations. "If two or more 
curves seem to give about equally good representations, it is 
quite possible to interpolate elements between the graphs 

plotted."f 

By this process values of e correct to within one- or two- 
hundredths and of w correct within a few degrees can generally 
be obtained at the first trial and with an expenditure of less 
than 10 minutes' time. The time of periastron passage fol- 
lows at once, and this set of preliminary elements may then 
be used as the basis for a least squares adjustment. The pro- 
cedure has been found very satisfactory at the Dominion Observa- 
tory at Ottawa, and at the Dominion Astrophysical Observatory, 
Victoria , B.C., where very many orbits of spectroscopic binary 
stars have been computed. 

* Since the velocity curve is ordinarily based on from fifteen to twenty 
normal places the work of multiplying by the appropriate reduction factors 
will require a very few minutes only. Of course, the amplitude of the curve 
as well as the period must be known before the reduction factors can be 
obtained. These are known with sufficient accuracy from the preliminary 
plots. 

tR. K. Young, Orbit of the Spectroscopic Binary 2 Sagittae, Jour. 
R.A.S.C. 11, 127, 1917. 



174 



THE BINARY STARS 



RUSSELL'S SHORT METHOD 

Professor Henry Norris Russell has devised a graphical 
method which is equally simple in its practical application.* 
Write Eq. (la) in the form 

p =V+^= V + Kecosa + Kcos (v + a) = 
at 

G + K cos (v + <o) (17) 

where p represents the observed radial velocity. 

Then (G + K) is the maximum, (G — K) the minimum 
value of the velocity, so that G and K may be estimated at 
once from the free-hand curve. The period is also assumed 
to be known. Equation (17) may then be written in the form 



cos {V + Oj) = — ~ — 



(18) 



and the value of (v + a>) computed for each observed value of p. 

If we subtract the corresponding values of M + Mo from each of 
these, we shall have values of (v — M ) + (w — M ). The second part 
of this expression is constant, while the first is the equation of the 
center in the elliptic motion. During a revolution this varies between 
equal positive and negative limits which depend only on the eccen- 
tricity, and are nearly proportioned to it, as is shown in the following 
table. 



Eccentricity 
Maximum equa- 
tion of center. 



0.10 



11?5 



0.20 



23?0 



0.30 



34?8 



0.40 

46?8 



0.50 
59?2 



0.60 
72?3 



0.70 
86?4 



0.80 
102?3 



0.90 

122?2 



If the values of (v — M) + (w — M ) are plotted against those of 
M + M o, we obtain a diagram which, since it represents the relations 
between the mean and the true anomalies, we may call the anomaly 
diagram. If on this diagram a curve is drawn to represent the plotted 
points, half the difference between its maximum and minimum ordi- 
nates will be the greatest value of the equatioD of the center, from which 
e may be found at once by means of the table. The mean of the maxi- 
mum and minimum ordinates will be the value of w - M . The 
instants when (v — M) -f- (co — Mo) has this value are those of periastron 
and apastron passage, the former corresponding to the ascending branch 
of the curve, which is always the steeper. The abscissae of the corre- 



Ap. Jour. 40, 282, 1914. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 175 

sponding points of the curve are M and M + 180°. The values of e, 
M , and co are now known, and the remaining elements may be found 
at once from K and G. 

„ According to Russell, the "principal advantage of this method 
is that the form of the curves which give v — M as a function 
of M depends upon e alone." For further details and an illus- 
trative example we refer the reader to the original memoir. Up 
to the present time the method has not come into general use. 
This remark applies also to the methods which have been 
developed by Laves, Henroteau, and more recent writers on the 
subject. References to the original papers for a number of these 
methods are given at the end of this chapter. 

DIFFERENTIAL CORRECTIONS TO THE ELEMENTS 
Whatever method may be used in finding the preliminary 
orbit it is generally advisable to determine the correction to the 
elements by the methods of least squares. * The formula derived 
by Lehmann-Filh6s from which the coefficients for the observa- 
tion equations are to be computed may be written as follows: 

S. = dV + [cos (v + «) + e cos w]dK 
at 

, r S sin (v + a?) sin v, n , N 1 , 

+K\ cos co v 1 _ \ (2 + e cos v) \de 

— i<L[sin (v + co) + e sin co]dco 

* Publ. Allegheny Obs. 1, 33, 1908. 
The advantages of applying the method of least squares to the definitive 
solution of spectroscopic binary stars have been clearly stated by Schlesinger 
in this paper. "The problem," he says, "involves the evaluation of five 
quantities (six if the period be included) that are so interwoven as to make 
their separate determination a matter of some difficulty. Herein lies the 
chief advantage for this case of the method of least squares; for it enables 
us to vary all of the unknowns simultaneously instead of one or two at a 
time. . . . Further, the method of least squares not only has the advantage 
of greater accuracy and of telling us how reliable our results are, but it 
eliminates from the computations any personal bias or arbitrary step . . . 
the method should be used in almost every case where the elements are not 
avowedly provisional." Not all computers are so enthusiastic as to the 
method. Judgment must of course be exercised in all orbit work as to 
whether the data at hand warrant anything beyond the computation of 
purely provisional elements. In spectroscopic binary orbits, for example, 
such factors, among others, as the number and quality of the plates and their 
distribution over the velocity curve, the character of the star's spectrum, and 
the character of the orbit must be considered in making this decision. 



176 THE BINARY STARS 

+ sin (v + «)(1 + e cos y)* _^ ' e% y flT 
- sin (v + «)(1 + e cos *>) 2 (1 5 e «) K« " ^ < 19 > 

In practice the period is almost always assumed to be known 
with accuracy and the last term of the equation is omitted. 

To facilitate the computation, Schlesinger has transformed 
this equation as follows: Put 

a = 0.452 sin v(2 + e cos v) 

_ (1 + e cos v) 2 
* (1 + e) 2 

T = dV + e cos w dK + K cos w de — Ke sin w dw 
k = dK 



T 



= -Kdu 



ir 2 - 21 j 

m = -K X L-^ • i dM, and w = (t> + «) 

\1 — e 1 — e 

Then the equations of condition take the form 

(& = r + cos u • k + sin u • T + a sin u • e 

+ p sin w • r + /3 sin w • (* - T)m (20) 

The quantities a and /3 can be tabulated once for all and such 
a tabulation is given by Schlesinger* so arranged "as to render 
the normal equations homogeneous and to enable all multiplica- 
tions to be made with Crelle's tables without interpolation." 
If this notation is used, the computer should have these tables 
at hand. 

When both spectra are visible on the plates, the orbits for 
the two components with respect to the center of mass may be 
determined separately. It is obvious that the two sets of 
values of V, e, T, and P must be identical, the values of u> must 
differ by 180°, while the two values for K depend upon the 
relative masses of the components. The preliminary elements 

* Loc. cU. 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 177 

for the two components, when independently determined, 
will, in general, not harmonize perfectly. To obtain the defini- 
tive values the best procedure is the one first suggested, I believe, 
by Dr. King.* It consists in combining all the observations, 
those for the secondary with those for the primary, into a single 
set of observation equations (equations of condition) and solving 
for one complete set of elements. If we write co' = (co + 180°) 
and distinguish the values of K for the two components by 
writing K x and K 2 , respectively, the equations in the notation 
of Eq. (19) assume the form 



d~jL = dV + [cos(t>+co) +■ e cos wJdKi+fcos^ + co')+e cosco'JciK^ 
at 

, ([ sin (v + co) sin v (c . . N ~| v 

+)\ cos co v ' 2 (2 + e cos i>) Ai 

. T / sin (v + co') sin v, n . ,~\ v \ , 

+ cos co' l - e 2 ( 2 + e cos w ) \ K *\ de 

— {[sin (v + co) + e sin co]Xi + [sin (v + co') + e sin co']X 2 }rfco 



+ [sin (v + co)(l + e cos t>) 2 i£i 

+ sin (v + co'Xl + e cos t>) 2 K 2 ] (1 * e ^ dT (21) 

the value of the period being assumed to require no correction. 

Since K 2 does not affect the residuals of the primary com- 
ponent, nor Ki those of the secondary, the terms containing 
dK 2 and dK x disappear from the equations representing the 
residuals from the primary and secondary curves, respectively. 

SECONDARY OSCILLATIONS 

When the orbit of a spectroscopic binary star has been com- 
puted and the theoretical velocity curve drawn, it is some- 
times found that the observed normal places are so distributed 
with respect to the curve representing simple elliptic motion 
as to suggest that a secondary oscillation is superimposed upon 
it. The question is whether this grouping arises from some 
source of error in the measurement of the spectrograms, from 

* See Harper's paper, in Publ. Dominion Obs., 1, 327, 1914. Dr. Paddock 
independently developed an equivalent equation. Lick Obs. Butt. 8, 166, 
1915. 



178 THE BINARY STARS 

erroneous values of one or more of the orbit elements, or from 
a real oscillation such as might be produced, for example, by 
the presence of a third body in the system. This question has 
been discussed by several investigators, among them Schlesinger, 
Zurhellen, and Paddock. As early as 1911, Schlesinger, then 
at the Allegheny Observatory, and his colleagues there, showed 
that the "blend effect" caused by the overlapping of the absorp- 
tion lines of the two component spectra "may produce such an 
apparent oscillation." They have also shown that it may be 
produced by chance errors in the velocities derived from the 
different lines of the spectrum, and they are convinced that a 
critical analysis of the data will dispose of a considerable per- 
centage of cases wherein secondary oscillations have been 
suspected. Later investigations by others have confirmed these 
conclusions. 

The possibility of a real secondary oscillation must, of course, 
be recognized, and when a full analysis has shown that such an 
oscillation is present, additional terms may be introduced into 
the equations of condition to represent it upon the assumption 
that it is produced by a third body revolving in a circular orbit 
about one of the other two components. In the cases that have 
arisen thus far this simple assumption has yielded a satisfactory 
representation of the data, though it is apparent that there is no 
reason for limiting such additional bodies to circular orbits. 

Let T' represent the time when the secondary curve crosses 
the primary from below, K' the semiamplitude of the sec- 
ondary oscillation, m' the ratio of the principal period to that 
of the secondary oscillation, assumed to be known (it is gen- 
erally taken to be an integer), and put u' — m'n(t - T'), 
t' = —m'uK'dT', k' — dK'; then the additional terms required 
in Eq. (20) are 

+ sin v! ■ k' + cos v! ■ r' (22) 

For a more complete discussion of secondary oscillations the 
reader is referred to the articles cited above. 

ORBITS WITH SMALL ECCENTRICITY 

In a circular orbit the elements T and « obviously have no 
significance, and when the eccentricity is very small, they 
become practically indeterminate by the geometrical or graph- 
ical methods which have here been described. Further, if 



THE ORBIT OF A SPECTROSCOPIC BINARY STAR 179 

approximate values are assumed, it is impossible to find cor- 
rections to both elements from the same least squares solution 
because the coefficients for the differential corrections will be 
nearly or quite equal. Some computers have overcome this 
difficulty by assuming the preliminary value of co as final, and 
deterinining corrections to T, but this is hardly a solution of 
the problem. In such orbits the analytic method possesses 
great advantages, as has been shown by several investigators, 
notably Wilsing, Russell, Zurhellen, and Plummer. Paddock 
has examined the question in great detail, extending some 
of the earlier developments and adapting them for computation. 
A full account of these methods would require more space than 
is available here, and it has seemed best to refer the reader to 
the original papers. 

References 

The following list of papers relating to one phase or another of the com- 
putation of orbits of spectroscopic binary stars, while not exhaustive, con- 
tains most of the more important ones. 

Rambaut, A. A.: On the Determination of Double Star Orbits from Spectro- 
scopic Observations of the Velocity in the Line of Sight, Mon. Not. 
R.A.S. 61, 316, 1891. 
Wilsing, J.: tlber die Bestimmung von Bahnelementen enger Doppel- 
sterne aus spectroskopischen Messungen der Geschwindigkeits-Com- 
ponenten, A.N. 134, 89, 1893. 
Lehmann-Filhbs, R.: tfber die Bestimmung einer Doppelsternbahn aus 
spectroskopischen Messungen der im Visionsradius liegenden Ge- 
schwindigkeits-Componente, A.N. 136, 17, 1894. 
Schwarzschild, K.: Ein Verfahren der Bahnbestimmung bei spectro- 
skopischen Doppelsternen, A.N. 152, 65, 1900. 
Russell, H. N.: An Improved Method of Calculating the Orbit of a Spectro- 
scopic Binary, Ap. Jour. 16, 252, 1902. 

: A Short Method for Determining the Orbit of a Spectroscopic 

Binary, Ap. Jour. 40, 282, 1914. 
Nltland, A. N.: Zur Bahnbestimmung von spektroskopischen Doppel- 
sternen, A.N. 161, 103, 1903. 
Laves, K.: A Graphic Determination of the Elements of the Orbits of 

Spectroscopic Binaries, Ap. Jour. 26, 164, 1907. 
Zurhellen, W. : Der spectroskopische Doppelstern o Leonis, A.N. 173, 353, 

1907. 
: Bemerkungen zur Bahnbestimmung spectroskopischer Doppelsterne, 

A.N. 176, 245, 1907. 

: Weitere Bemerkungen zur Bahnbestimmung spectroskopischer 

Doppelsterne, u.s.w., A.N. 177, 321, 1908. 

t)ber sekondare Wellen in den Geschwindigkeits-Kurven spectro- 



skopischer Doppelsterne, A.N. 187, 433, 1911. 



180 THE BINARY STARS 

King, W. F.: Determination of the Orbits of Spectroscopic Binaries, Ap. 
Jour. 27, 125, 1908. 

Curtis, H. D.: Methods of Determining the Orbits of Spectroscopic Bin- 
aries, Publ. A.S.P., 20, 133, 1908. (This paper has, with the author's 
permission, been very freely used in preparing my chapter on the 
subject.) 

Plttmmer, H. C. : Notes on the Determination of the Orbits of Spectroscopic 
Binaries, Ap. Jour. 28, 212, 1908. 

Schlbsinger, F.: The Determination of the Orbit of a Spectroscopic 
Binary by the Method of Least Squares, Publ. Allegheny Obs. 1, 33, 1908. 

: On the Presence of a Secondary Oscillation in the Orbit of 30 H 

Ursae Majoris, Publ. Allegheny Obs. 2, 139, 1911. 
-: A Criterion for Spectroscopic Binaries, etc., Ap. Jour. 41, 162, 1915. 



Paddock, G. F.: Spectroscopic Orbit Formulae for Single and Double 

Spectra and Small Eccentricity, Lick Obs. Bull. 8, 153, 1915. 
Cttrtiss, R. H. : Method of Determining Elements of Spectroscopic Binaries, 

Publ. Astron. Obs. Univ. of Michigan 2, 178, 1916. 
Henroteatt, F. : Two Short Methods for Computing the Orbit of a Spectro- 
scopic Binary Star by Using the Allegheny Tables of Anomalies, Publ. 

A.S.P. 29, 195, 1917. 
King, E. S. : Standard Velocity Curves for Spectroscopic Binaries, Harvard 

Ann. 81, 231, 1921. 
Halm, J. K. E. : On a Graphical Determination of the Orbital Elements of a 

Spectroscopic Binary, Mon. Not. R.A.S. 87, 628, 1927. 
Picart, L. : B^emarques sur le calcul des orbites des 6toiles doubles spectro- 

scopiques, J.O. 10, 137, 1927. 
Pogo, A. : On the Use of the Hodographic Method of Laves for Determining 

Elements of Spectroscopic Orbits, Ap. Jour. 67, 262, 1928. 
Maderni, A.: La determinazione degli elementi orbitali di una doppia 

spettroscopica, Mem. Soc. Astron. Italiana N.S. 5, 65, 1930. 
Orlopp, A. : Harmonic Tables for Spectroscopic Binaries, Odessa Astron. Obs. 

1930. 
Luyten, W. J.: A Rediscussion of the Orbits of 77 Spectroscopic Binaries (of 

Small Eccentricities), Ap. Jour. 84 (1), 85-103, July 1936. 

In his Third Catalogue of Spectroscopic Binary Stars (Lick Obs. Bull. 11, 
141, 1924) Dr. J. H. Moore lists, in separate tables, all stars for 
which a variation in radial velocity had been fairly established, 
before 1924.0. See also his Fourth Catalogue of Spectroscopic Binary 
Stars (Lick Obs. Butt. 18, 1-38, #483, 1936). 



CHAPTER VII 

ECLIPSING BINARY STARS 

We have seen that one of the first binary systems to be dis- 
covered with the spectrograph was Algol (0 Persei), long known 
as a variable star. There are other stars whose light varies in 
the same peculiar manner as that of Algol; that is, while it 
remains sensibly constant at full brightness the greater part 
of the time, at regular intervals it fades more or less rapidly 
to a certain minimum. It may remain constant at this mini- 
mum for a short time and then recover full brightness, or the 
change may be continuous. In either case the entire cycle of 
change is completed in a small fraction of the time of constant 
light between the successive niinima. 

The hypothesis that in every such case the star, as viewed 
from the Earth, undergoes a total, annular, or partial eclipse, 
the eclipsing body being a relatively dark star revolving with 
the other about a common center of gravity, completely accounts 
for the observed facts and has been proved to be correct not 
only in the one instance, Algol, but also in that of every Algol- 
type variable which has been investigated with the spectrograph. 
Undoubtedly it is the correct explanation for all stars of this 
type ; they are all binary systems. 

Unless the darker star is absolutely nonluminous, there 
should be a second ininimum when the bright star passes 
between it and the Earth, the relative depth of the two minima 
depending upon the relative intensity of the light of the two 
stars and upon their relative areas. Such a secondary mini- 
mum has been observed in /3 Lyrae and in this star the light 
is not quite constant at any phase, either maximum or min- 
imum. There is now no doubt but that this star and others 
like it are also binary systems. 

It was formerly thought that a distinction could be drawn 
between variable stars of the type of Algol and those of the 
type of j8 byrae, but measures with sensitive modern photom- 
eters, such as the selenium cell, the photoelectric cell, and 

181 



182 THE BINARY STARS 

the sliding-prism polarizing photometers, and measures of 
extrafocal images on photographic plates have attained such 
a degree of accuracy that a variation considerably less than 
one-tenth of a magnitude can be detected with certainty; and 
it now appears that Algol itself not only has a slight secondary 
minimum but that its light is nqt quite constant at maximum. 
The distinction, therefore, breaks down and we may regard 
all the stars of these two types as members of a single class, 
calling them eclipsing binaries or eclipsing variables, according 
to the point of view from which we take up their investigation. 
In all, nearly 1,000 eclipsing binary stars are known at the 
present time, a large percentage of them being too faint to 
photograph with our present spectrographic equipment. It 
is therefore a matter of great interest to inquire what informa- 
tion, if any, as to the orbits of these systems can be derived 
from their light curves, the curves, that is, which are constructed 
by taking the observed stellar magnitudes as ordinates and 
the corresponding times as abscissae. 

Up to the beginning of the present century observations of 
eclipsing variables were made chiefly to determine the times of 
minima accurately, with the object of improving the light ele- 
ments. As data accumulated, variations in the period began 
to be noted in many instances. Dr. Seth Chandler in particular 
was greatly interested in these variations and added parabolic 
or periodic terms to the light elements in his catalogues of variable 
stars published in the Astronomical Journal, but did not find an 
explanation for them. For the most part, in fact, these varia- 
tions are so complicated that they are still a puzzle in celestial 
mechanics, but in the case of Y Cygni* the variation has been 
definitely traced to a revolution of the line of apsides, produced, 
apparently, by the ellipsoidal figure of the component stars. 

The earlier observations, confined as they were chiefly to 
estimates of the time of minimum, were unsatisfactory material 
for the determination of the orbital elements. Among the 
pioneers in the accurate observation of the entire light curve 
we may name Roberts and Nijland, using the method of estimates 
developed by Argelander; Wendell and Dugan, using a polarizing 
photometer devised by E. C. Pickering; Stebbins, using selenium 
and photoelectric cells; and Parkhurst, Seares, and Baker, using 
photographic methods. 

♦Dugan, Contr. Princeton Univ. Obs. 12, 1931. 



ECLIPSING BINARY STARS 183 

Professor E. C. Pickering* made an investigation of the 
orbit of Algol on the basis of its light curve as early as 1880, 
and showed that a solution of such orbits was possible if certain 
reasonable assumptions— for example, that the two stars are 
spherical with uniformly illuminated disks and move in circular 
orbits — were granted. The subject was resumed by him later, 
and was taken up also by Harting, Tisserand, A. W. Roberts, 
and others. 

In the years 1912 to 1915, Russell and Shapley, at Princeton 
University, f made a very thorough investigation of the problem, 
Russell developing a general analytical method, which has 
formed the basis of nearly all later calculations, and Shapley 
applying it to the computations of the orbit elements of 90 sys- 
tems, 31 of the solutions being indicated as of the first grade. 
Modifications of this method have been proposed by various 
investigators, and Sitterly has developed a graphical method, 
but in the present chapter Russell's method will be given. 

In the most general case the problem is an extremely com- 
plicated one, for the orbits must be regarded as elliptical with 
planes inclined at a greater or less angle to the line of sight, 
and the two components as ellipsoids, the longest diameter of 
each being directed toward the other star. Moreover, the 
disks may or may not be uniformly illuminated; they may be 
darker toward the limb, as our own Sun is, the degree of dark- 
ening depending upon the depth and the composition of the 
enveloping atmosphere; and the side of each which receives 
the radiation from the other may be brighter than the opposite 
side. The complete specification of an eclipsing binary system 
therefore requires a knowledge of at least 13 quantities which, 
in Russell's notation, are as shown in the table on page 184. 

The longitude of the node must remain unknown, as there is no hope of 
telescopic separation of any eclipsing pair. 

The value of a in absolute units can be found only from spectroscopic 
data. In the absence of these, it is desirable to take a as an unknown 
but definite unit of length, and express all other linear dimensions in 
terms of it. Similarly, the absolute values of L x and L 2 can be deter- 
mined only if the parallax of the system is known. But in all cases the 

* Dimensions of the Fixed Stars, with special reference to Binaries and 
Variables of the Algol Type, Proc. Amer. Acad. Arts and Sciences 16, 1, 1881. 

t References to their papers are given, with others, at the end of this 
chapter. 



184 



THE BINARY STARS 



combined light of the pair, L x + L 2 , can be taken as the unit of light 
and the apparent brightness at any time expressed in terms of this. 
This leaves the problem with eleven unknown quantities to be deter- 
mined from the photometric measures. Of these, the period is invari- 
ably known with a degree of accuracy greatly surpassing that attainable 
for any of the other elements, and the epoch of principal minimum can 
be determined, almost independently of the other elements, by inspec- 
tion of the light-curve. Of the remaining elements, the constants 
expressing ellipticity and reflection may be derived from the observed 
brightness between eclipses. These effects are often so small as to be 
detected only by the most refined observations. The question of 
darkening toward the limb may well be set aside until the problem is 
solved for the case of stars that appear as uniformly illuminated disks. 



Orbital Elements 



Semimajor axis 

Eccentricity 

Longitude of periastron 

Inclination 

Period 

Epoch of principal conjunction 



Eclipse Elements 



Radius of larger star 

Radius of smaller star 

Light of larger star 

Light of smaller star 

and at least 3 constants defining 
the amount of elongation, of 
darkening at the limb, and of 
brightening of one star by the 
radiation of the other. 



U 



This leaves us with six unknowns. Fortunately, systems of such 
short periods as those of the majority of eclipsing variables, usually have 
nearly circular orbits (as is shown both by spectroscopic data and by the 
position of the secondary minimum). The assumption of a circular 
orbit is therefore usually a good approximation to the facts and often 
requires no subsequent modification. 

Russell's papers discuss first the simplified problem: 

Two spherical stars, appearing as uniformly illuminated disks, and 
revolving about their common center of gravity in circular orbits, mutually 
eclipse one another. It is required to find the relative dimensions and 
brightness of the two stars, and the inclination of the orbit, from the observed 
light curve. 

The determination of the orbit can be made by simple geometrical 
methods, but their practical application demands the tabulation and 
use of rather complicated functions. 

We may assume P and t as already known. If the radius of the rela- 
tive orbit is taken as the unit of length, and the combined light of the 



ECLIPSING BINARY STARS 185 

two stars as the unit of light, we have to determine four unknown 
quantities. Of the various possible sets of unknowns, we select the 
following: 

Radius of the larger star r i 

Ratio of radii of the two stars * 

Light of the larger star Li 

Inclination of the orbit * 

The radius of the smaller star is then r 2 = kr h and its light, L 2 = 1 — L\. 
It should be noticed that, with the above definitions, k can never exceed 
unity, but L 2 will exceed L x whenever the smaller star is the brighter 
(which seems to be the fact in the majority of observed cases). 

The development of the subject that follows is given, by his 
courteous permission, largely in Russell's own words, taken 
partly from his printed memoir but chiefly from a summary 
sent me recently in manuscript form. The numbers for the 
equations below are those given in his original paper. 

The simplest case is that of a total eclipse which can often be recog- 
nized on inspection by a deep, fiat-bottomed minimum in which the 
light is Lx or X. Then L 2 = 1 - X. If the loss of light at any moment 
during the partial phase is ctL 2 the fraction a of the disk of the smaller 
star must be obscured. By geometrical similitude a depends only on 
the ratio k of the radii of the disks and the ratio 5/n of their projected 
distance of centers to the larger radius, and we may write 

«->(*■£) 

The function / is transcendental but may be computed with ordinary 
trigonometric tables. For any given value of k, we may invert this 
function, and set 

- = <K&, a) (9) 

If is the longitude in the orbit, measured from mideclipse, 

S 2 = cos 2 i + sin 2 i sin 2 = r^l/pffc, «)] 2 (11) 

Now let a u a 2 , a% be any definite values of a and d u 2 , 03 the cor- 
responding values of (which may be found from the light curve). 
Subtracting the corresponding equations of the form (11) in pairs, and 
dividing one of the resulting equations by the other, we find 

sin 2 0i - sin 2 2 \4>(k, «i)] 2 - [<t>(k, <* 2 )] 2 _ ... v (12) 

sin 2 2 - sin 2 03 = [0(fc, « 2 )1 2 - [<t>(k, «a)T 2 ~ ^' " h "* °* J K > 



186 THE BINARY STARS 

The first member of this equation contains only known quantities. 
The second, if ct\, at, and a 3 are predetermined, is a function of k alone. 
If this function is tabulated, the value of k in any given case can be 
found by interpolation, or graphically. Equation (11) can then be 
used to find n and i. 

A theoretical light curve may then be found, which passes through 
any three desired points on each branch of the observed curve (assumed 
symmetrical). These points may be chosen at will by altering the 
values of «i, a 2 , and a 3 . In practice it is convenient to keep a 2 and a 3 
fixed, so that \f/ becomes a function of k and a\ only, and may be tabu- 
lated for suitable intervals in these two arguments. This has been 
done in Table II, in which a 2 is taken as 0.6 and a 3 as 0.9. If A — sin 2 2 , 
B = sin 2 6 2 — sin 2 6 3 , (12) may be written 

sin 2 0i = A + B^(k, ttl ) (13) 

The points a and & on the light curve corresponding to a 2 and a 3 , together 
with the point corresponding to any one of the tabular values of ai, then 
give a determination of k. By taking a suitably weighted mean of 
these values of k, a theoretical light curve can be obtained which passes 
through the points a and 6, and as close as possible to the others. By 
slight changes in the assumed positions of a and b (i.e., in the corre- 
sponding values of 6, or of t — t ), it is possible with little labor to obtain 
a theoretical curve which fits the whole course of the observed curve 
almost as well as one determined by least squares. The criterion of 
this is that the parts of the observed curve below b (near totality), 
between a and b, and above a (near the beginning or end of eclipse) 
shall give the same mean value of k. The individual determinations 
of k are of very different weight. Between a and 6 (that is for values of 
a\ between 0.6 and 0.9) \f/ changes very slowly with k. At the beginning 
and end of the eclipse the stellar magnitude changes very slowly with 
the time, and hence, by (13), with yj/. The corresponding parts of the 
curve are therefore ill adapted to determine k. For the first approxi- 
mation it is well to give the values of k derived from values of «i between 
0.95 and 0.99, and between 0.4 and 0.2, double weight (provided the 
corresponding parts of the curve are well fixed by observation). The 
time of beginning or end of eclipse cannot be read with even approxi- 
mate accuracy from the observed curve and should not be used at all in 
finding k. The beginning or end of totality may sometimes be deter- 
mined with fair precision, but does not deserve as much weight as the 
neighboring points on the steep part of the curve. If further refinement 
is desired, it can most easily be obtained by plotting the light curve for 
two values of k and comparing with a plot of the observations. This 
will rarely be necessary. 

When once k is given, the determination of the light curve is a very 
easy matter. For each tabular value of a t , Eq. (13) gives 6 U and hence 



ECLIPSING BINARY STARS 187 

(h - t ). The values of the stellar magnitude m corresponding to given 
values of «i are already available, having been used in the previous work. 
The light curve may thus be plotted by points in a few minutes. 

After a satisfactory light curve has been computed, we may proceed 
to determine the remaining elements. Let d' and 6" be the values 
corresponding to the beginning of eclipse («i = 0) and to the beginning 
of totality (ai = 1). Then by Eq. (13) 

sin 2 d' = A + B+(k, 0) and sin 2 d" = A + B+(k, 1) 

These computed values are more accurate than those estimated from 
the free-hand curve drawn to represent the observations. At the first 
of these epochs 5 = n + r 2 , and at the second 5 = n - r 2 . We have 
then, by Eq. (11) 

ri 2(i + jfe)2 = cos 2 i + sin 2 i sin 2 d' 
ri 2 (l - kY = cos 2 i + sin 2 i sin 2 d" 

whence 

4k cot 2 % = (1 - fc) 2 sin 2 6' - (1+ *) a sin 8 0" 
4Ati 2 (1 + cot 2 i) = sin 2 & - sin 2 6" 

Introducing A and B, we have 

4fc cot 2 i = -MA + B[(l - *)V(*. 0) - (1 + *)V(*. 1)1 
4AT! 2 cosec 2 i = B[^(k, 0) - M, 1)] 

The coefficients are functions of k alone, and may be tabulated. It is 
most convenient for this purpose to put the equations in the form 

B 

rS cosec 2 1 = ^ ^ ^ 

COt2l ' = *ffc)- A 

as in this way we obtain functions whose tabular differences are com- 
paratively smooth (which is not true of their reciprocals). With the 
aid of these functions the elements may be found as soon as A and B 
are known. 

Occasionally cot 2 i comes out negative. The curve must then 
be fitted as well as possible on the assumption of central transit 

(cot i = 0). 

The secondary eclipse will then be annular, with maximum 
depth 1 - X = fc 2 Li. When observed, it may be used to test 
the applicability of the simplified theory. In a few cases the 



188 THE BINARY STARS 

eclipse at principal minimum may be annular. The process of 
solution is identical, but the computed depth of secondary is 
greater. If (1 — X) > k 2 only the total solution is possible, 
since h\ cannot exceed 1. Otherwise the secondary minimum 
must be observed to settle the question. When the eclipses 
are partial (round-bottomed curves) an additional unknown 
has to be found, «o, the maximum fraction of the area of the 
smaller star which is eclipsed. The detailed analysis shows that 
if only the primary minimum has been observed, a variety of 
solutions, with different values of k and a , can be found, which 
will give light curves practically indistinguishable even by the 
best observations. The computed depth of secondary minimum, 
however, differs from one solution to another ; and if this has been 
observed, a definite determination is possible so long as the 
primary is deep — or both eclipses fairly deep. When both 
are shallow there is a considerable range of admissible solutions, 
and a definite answer can be obtained only from additional 
data, such as a spectrographs estimate of the relative brightness 
of the components. 

Russell's paper contains a number of tables, of which the 
four directly applicable to the cases of total or annular eclipses 
which have been discussed are reprinted here. His Table I, 
tabulating the function given in Eq. (9), is omitted, though it is 
fundamental, because it is used, so far as we are at present 
concerned, only in constructing the subsequent tables. 

His Table II contains the function ^(fc, a x ) defined by the equa- 
tion 

ith ^ - [1 + ^ (fc > ai)]2 ~ [1 + k P (k > " 2 > ]2 
* K ' ai) " [1 + *p(*, « 2 )] 2 - [1 + k V {k, a 3 )] 2 

(where a 2 = 0.6 and a* = 0.9), which is used in determining k 
in the case of total eclipse. The uncertainty of the tabular quan- 
tities does not exceed one or two units of the last decimal place, 
except for the larger values of xf/, corresponding to values of a\ 
less than 0.3, for which the actual errors may be greater, but 
are not more serious in proportion to the whole quantity 
tabulated. 
Table Ha contains the functions 

<t>i(k) = 



*(fc, 0) - *(fc, 1) 



ECLIPSING BINARY STARS 



189 



o 

T-H 

© 



o 
oo 
H 

P 



OQ 

CM 

3 



o 

M 

OQ 

« 

o 



3 

d 



S§i83aisSS38S§a8388§agSS 

^^^--00 = 00 ooooooo----- 

Jasgassselssssssssssssss 

^ei ^ ^ ^ ^ ^ d d d d o o o o o o o o ^ - - - ^ 

853883883888288858888838 

C, N CH.HHHOOOOOOOOOOOHHHHH 

sisaassssBiisissisaiSiM. 

„ NN? HHHHOOOOOOOOOOOHHHHH 

SS&S8S5l8SfcSSoS8^S5S5Ssi:Sa5 

^ei««HHHOJed ? OOOOOOHHHHH 

iS3B88S3g»88S§S88iaiS«B? 

^•««HHHHOOOOooeeeoHHHHH 

5§S8S§I3S5i3a§Si8IS§aS8S 

4. -j- + + + I I I • I I ' ' ' ' 

»^« N ciHHHOOOOOOOOOOHHHHH 

aBS8aaaa»8ES88asaBB83E8a 

553B83S833S38S833M.383M 

_ — WMQ 

3gS8S8SgS3SSS8S5sS88S88S 

dx^g^gc^^dooooooo^^^ 



|QO>00!0 



I© 



IOQOAO 

ddddddddddddooooooooooo^H 



8§§2^§SSS§^^SSSoS?:S5««*o5« 



190 

and 



<t>*(k) = 



THE BINARY STARS 
4& 



(l - *)V(*, 0) - (l + *)V(*, i) 

which are useful in determining the elements in the case of total 
eclipse. 

Table Ila. — For Computing the Elements in the Case op Total 

Eclipse 



k 


*i(*0 


fa(k) 


1.00 


0.380 


0.939 


0.95 


0.401 


0.894 


0.90 


0.417 


0.848 


0.85 


0.427 


0.802 


0.80 


0.431 


0.755 


0.75 


0.431 


0.709 


0.70 


0.427 


0.663 


0.65 


0.419 


0.617 


0.60 


0.406 


0.572 


0.55 


0.390 


0.527 


0.50 


0.371 


0.482 


0.45 


0.349 


0.436 


0.40 


0.323 


0.390 


0.35 


0.294 


0.345 


0.30 


0.262 


0.298 


0.25 


0.226 


0.250 


0.20 


0.187 


0.202 


0.15 


0.145 


0.153 


0.10 


0.100 


0.103 


0.05 


0.052 


0.052 


0.00 


0.000 


0.000 



Table A gives the loss of light (1 — X), corresponding to a 
given change Aw in stellar magnitude. For a difference of 
magnitude greater than 2.5, the loss of light is 0.9000 + one- 
tenth of the tabular value for Am — 2™5. Table B gives the 
values of (0 — sin 0) for every 0.01 of (expressed in circular 
measure), and saves much labor in computing the values of sin 
corresponding to a given interval for minimum. 



ECLIPSING BINARY STARS 



191 



Table A.— Loss of Light Corresponding to an Incbease Am in Stellab 

Magnitude 



o.o 

0.1 
0.2 
0.3 
0.4 

0.5 
0.6 
0.7 
0.8 
0.9 

1.0 
1.1 
1.2 
1.3 
1.4 

1.5 
1.6 
1.7 
1.8 
1.9 

2.0 
2.1 
2.2 
2.3 
2.4 

2.5 



0.0000 
0.0880 
0.1682 
0.2414 
0.3082 

0.3690 
0.4246 
0.4752 
0.5214 
0.5635 

0.6019 
0.6369 
0.6689 
0.6980 
0.7246 

0.7488 
0.7709 
0.7911 
0.8095 
0.8262 

0.8415 
0.8555 
0.8682 
0.8798 
0.8904 



0.0092 
0.0964 
0.1759 
0.2484 
0.3145 

0.3748 
0.4298 
0.4800 
0.5258 
0.5675 

0.6055 
0.6403 
0.6719 



0.0183 
0.1046 
0.1834 
0.2553 
0.3208 

0.3806 
0.4351 
0.4848 
0.5301 
0.5715 

0.6092 
0.6435 
0.6749 



0.7008 0.7035 



0.72710.7296 0.7321 



0.0273 
0.1128 
0.1909 
0.2621 
0.3270 

0.3862 
0.4402 
0.4895 
0.5344 



0.0362 
0.1210 
0.1983 
0.2689 
0.3332 



0.0450 
0.1290 
0.2057 
0.2756 
0.3393 



0.3919 0.3974 
0.4454 0.4505 



0.4942 
0.5387 
0.5754 0.5793 



0.4988 



0.6127 
0.6468 
0.6779 
0.7062 



0.7511 
0.7730 
0.7930 
0.8112 
0.8278 

0.8430 
. 8568 
0.8694 
0.8809 
0.8914 



0.9000 0.9009 



0.7534 
0.7751 
0.7949 
0.8129 
0.8294 

0.8444 
0.8581 
0.8706 
0.8820 
0.8924 



0.7557 
0.7772 
0.7968 
0.8146 
0.8310 



0.0538 
0.1370 
0.2130 
0.2822 
0.3454 

0.4030 
0.4555 
0.5034 



0.5429 0.5471 
0.5831 0.5870 



0.6163 
0.6501 
0.6808 
0.7089 
0.7345 

0.7579 
0.7792 
0.7986 
0.8163 
0.8325 



0.6198 
0.6533 
0.6838 
0.7116 



0.7370 0.7394 0.7418 



0.8458 0.8472 



0.8594 
0.8718 
0.8831 
0.8933 



0.9018 0.9027 



0.8607 
0.8729 
0.8841 
0.8943 



8 9 



0.6233 
0.6564 
0.6867 
0.7142 



0.0624 
0.1449 
0.2202 
0.2888 
0.3514 

0.4084 
0.4605 
0.5080 
0.5513 
0.5907 

0.6267 
0.6596 
0.6895 
0.7169 



0.7601 
0.7812 
0.8005 
0.8180 
0.8340 

0.8486 
0.8620 
0.8741 
0.8852 
0.8953 



0.7623 
0.7832 
0.8023 
0.8197 
0.8356 

0.8500 
0.8632 
0.8753 
0.8862 
0.8962 



0.9036 0.9045 0.9054 



0.0710 
0.1528 
0.2273 
0.2953 
0.3573 

0.4139 
0.4654 
0.5125 
0.5554 
0.5945 

0.6302 
0.6627 
0.6924 
0.7195 
0.7441 



0.0795 
0.1605 
0.2344 
0.3018 
0.3632 

0.4192 
0.4703 
0.5169 
0.5594 
0.5982 

0.6336 
0.6658 
0.6952 
0.7220 
0.7465 



0.7645 
0.7852 
0.8041 
0.8214 
0.8371 

0.8514 
0.8645 
0.8764 
0.8873 
0.8972 

0.9062 



0.7667 
0.7872 
0.8059 
0.8230 
0.8386 

0.8528 
0.8657 
0.8775 
0.8883 
0.8981 

0.9071 



0.7688 
0.7891 
0.8077 
0.8246 
0.8400 

0.8541 
0.8670 
0.8787 
0.8893 
0.8991 

0.9080 



For values of Am greater than 2.5, the loss of light is 0.9000 plus Ho of the loss of light 
corresponding to Am — 2.5. 

Table B. — Values of — sin 





0.0 


0.1 


0.2 


0.3 


0.4 


0.5 


0.6 


0.7 


0.8 


0.9 


0.00 


0.0000 


0.0002 


0.0013 


0.0045 


0.0105 


0.0206 


0.0354 


0.0558 


0.0826 


0.1167 


0.01 


0.0000 


0.0002 


0.0015 


0.0049 


0.0114 


0.0218 


0.0372 


0.0582 


0.0857 


0.1205 


0.02 


0.0000 


0.0003 


0.0018 


0.0055 


0.0122 


0.0231 


0.0390 


0.0607 


0.0889 


0.1243 


0.03 


0.0000 


0.0004 


0.0020 


0.0060 


0.0131 


0.0244 


0.0409 


0.0632 


0.0920 


0.1283 


0.04 


0.0000 


0.0005 


0.0023 


0.0066 


0.0141 


0.0258 


0.0428 


0.0658 


0.0953 


0.1324 


0.05 


0.0000 


0.0006 


0.0026 


0.0071 


0.0151 


0.0273 


0.0448 


0.0684 


0.0987 


0.1365 


0.06 


0.0000 


0.0007 


0.0029 


0.0078 


0.0161 


0.0288 


0.0469 


0.0711 


0.1022 


0.1407 


0.07 


0.0001 


0.0008 


0.0033 


0.0084 


0.0171 


0.0304 


0.0490 


0.0739 


0.1057 


0.1450 


0.08 


0.0001 


0.0010 


0.0037 


0.0091 


0.0183 


0.0320 0.0512 


0.0767 


0.1093 


0.1494 


0.09 


0.0001 


0.0011 


0.0041 


0.0098 


0.0194 0.0337 JO. 0535 


0.0796 


0.1130 


0.1539 



192 



THE BINARY STARS 



To illustrate Russell's method I have chosen his orbit of 
W Delphini, which is a "typical Algol variable with a deep 
primary minimum, showing a constant phase, and little or no 
secondary minimum." Its light curve, "denned by the 500 
observations by Professor Wendell, with a polarizing photom- 
eter, which are published in the Harvard Annals, 69, Part 1," 

Table o. — Observed Magnitudes 



Phase 


Mag. 


No. 
obs. 


O -C 


Phase 


Mag. 


No. 
obs. 


O -C 


-0^2894 


9.41 


6 


+0"?01 


+0^0560 


11.76 


7 


+0"?01 


0.2637 


9.49 


5 


+0.02 


0.0659 


11.58 


8 


+0.01 


0.2458 


9.58 


5 


+0.04 


0.0753 


11.33 


7 


-0 .04 


0.2306 


9.59 


4 


-0.01 


0.0859 


11.14 


5 


-0.02 


0.2200 


9.67 


5 


0.00 


0.0937 


10.97 


5 


-0.05 


0.2106 


9.73 


8 


+0.01 


0.1036 


10.88 


8 


+0.02 


0.2007 


9.79 


10 


0.00 


0.1147 


10.73 


8 


+0.05 


0.1911 


9.88 


12 


+0.02 


0.1246 


10.56 


12 


+0.03 


0.1817 


9.95 


10 


+0 .01 


0.1351 


10.39 


14 


0.00 


0.1718 


10.02 


8 


.00 


0.1445 


10.31 


11 


+0.04 


0.1615 


10.16 


17 


+0.04 


0.1546 


10.13 


10 


-0.02 


0.1506 


10.23 


14 


0.00 


0.1641 


10.10 


11 


+0.04 


0.1396 


10.37 


14 


+0.01 


0.1744 


9.97 


10 


0.00 


0.1311 


10.44 


16 


-0.03 


0.1847 


9.90 


9 


+0.02 


0.1212 


10.59 


17 


-0.03 


0.1941 


9.79 


9 


-0.02 


0.1121 


10.78 


14 


+0.01 


0.2050 


9.71 


8 


-0.02 


0.1013 


10.91 


17 


-0.04 


0.2157 


9.71 


6 


+0 .04 


0.0906 


11.12 


14 


-0.01 


0.2242 


9.63 


8 


+0.01 


0.0809 


11.30 


10 


-0 .02 


0.2345 


9.57 


7 


0.00 


0.0715 


11.51 


12 


0.00 


0.2507 


9.50 


7 


0.00 


0.0617 


11.69 


10 


.00 


0.2708 


9.48 


7 


+0.03 


0.0509 


11.88 


7 


0.00 


0.2811 


9.43 


4 


+0.02 


0.0313 


12.05 


5 


-0 .04 


0.94 


9.42 


5 


+0.02 


0.0169 


12.08 


4 


-0 .02 


1.90 


9.35 


5 


-0.05 


-0.0082 


12.07 


7 


-0.03 


2.04 


9.41 


7 


+0.01 


+0.0060 


12.16 


5 


+0.06 


2.67 


9.38 


5 


-0.02 


0.0139 


12.09 


4 


-0 .01 


3.04 


9.42 


3 


+0.02 


0.0261 


12.03 


5 


-0.07 


4.04 


9.44 


6 


+0 .04 


0.0356 


12.02 


6 


-0 .03 


4.48 


9.36 


7 


-0.04 


+0.0460 


11.87 


6 


-0.04 











is shown in Fig. 12. The observations have been combined 
into the normal places given in Table o, on the basis of a period 
of 4.8061 days, which was found to require no correction. 



ECLIPSING BINARY STARS 



193 



From the 38 observations outside minimum we find the magnitude 
during constant light to be 9^395 + 0.009. There is no evidence of 
any change during this time. With a circular orbit, the secondary 
minimum should occur at phase 2?40. As none of the observations fall 
within 0?27 of this, they give us no information whether such a minimum 
exists. The light curve of the principal minimum is very well deter- 
mined. The eclipse lasts from about -0?28 to +0?28, and there is a 
short constant period at the middle, of apparently a little less than one- 
tenth the total duration of the eclipse. The mean of the 20 observa- 
tions lying within 0-02 of the middle of eclipse gives for the magnitude 
at this phase 12^10 ± 0.014. The range of variation is therefore 2?70, 

9^0 



lo'no 



llTO 



12T»0 



-0$3 "0^2 -OSll ($0 +0?1 +0?2 +0?3 

Fig. 12. — Light curve of the principal minimum of W Delphini. 

and the light intensity at minimum 0.0832 times that at maximum. 
This shows at once that the eclipse is total, for if it was annular, the 
companion (even if perfectly dark) must cut off at least 0.917 of the 
light of the primary, and hence its radius cannot be less than 0.956 times 
that of the latter. In such a system the duration of the annular phase 
could not exceed 0.044/1.956, or 0.022 of the whole duration of eclipse. 
The observed constant phase is almost five times as long as this. 

The brighter star, therefore, gives 0.9168 of the whole light of the 
system, and if isolated would appear of magnitude 9.49; while the 
fainter but larger star which eclipses it gives out only one-eleventh as 
much light, and when seen alone at minimum is of magnitude 12.10. 

The loss of light (1 - I) at any given time, t, will be 0.9168a!, 
since a x is the percentage of obscuration. For a series of values 
of a x we tabulate the values of (1 - I) and then take from 
Table A the corresponding changes of magnitude and apply 
them to magnitude 9.4. Next, from the free-hand curve drawn 




194 



THE BINARY STARS 



to represent the data of observation, we read off the epochs 
h and U at which the magnitudes so computed are reached 
before and after the middle of eclipse. Half the difference 
of t\ and t 2 may be taken as the interval t from the middle of 
eclipse to each phase and the corresponding value of 6 formed 
from 

= -pt = 1.3065J, where is expressed in radians and t in days. 



With the aid of Table B sin 6 is found and then sin 2 0. 
quantities are all entered in Table b. 

Table 6 



These 



a\ 


1 - I 


Mag. 


d 


h 





sin* 


sin 2 - A 


* (A, ai) 


k 


0.0 


0.0000 


9°400 


-04304: 


+04300: 


0.394: 


0.1474: 


0.1105: 


+4.28: 


0.56: 


0.1 


0.0917 


9 .505 


0.2540 


0.2515 


0.3304 


0.1050 


0.0681 


2.64 


0.504 


0.2 


0.1834 


9 .620 


0.2286 


0.2258 


0.2968 


0.0860 


0.0491 


1.908 


0.505 


0.3 


0.2760 


9.749 


0.2075 


0.2030 


0.2681 


0.0702 


0.0333 


1.290 


0.480 


0.4 


0.3667 


9.896 


0.1884 


0.1830 


0.2426 


0.0576 


0.0207 


0.802 


0.462 


0.5 


0.4584 


10 .066 


0.1682 


0.1644 


0.2173 


0.0462 


0.0093 


+0.361 


0.36: 


0.6 


0.5500 
0.6417 


10 .266 
10 .514 


0.1486 
0.1270 


0.1470 
0.1274 


0.1931 
0.1661 


0.0369 
0.0272 


0.0000 
-0.0097 






0.7 


-0.376 


0.64: 


0.8 


0.7334 


10 .835 


0.1070 


0.1048 


0.1381 


0.0190 


-0.0179 


-0.694 


0.56: 


0.9 


0.8250 


11 .292 


0.0824 


0.0788 


0.1054 


0.0111 


-0.0258 


-1.000 




0.9S 


0.8709 


11 .624 


0.0655 


0.0624 


0.0886 


0.0071 


-0.0298 


-1.156 


0.58: 


0.98 


0.8986 


11 .884 


0.0505 


0.0462 


0.0632 


0.0040 


-0.0329 


-1.277 


0.525 


0.99 


0.9076 


11 .985 


0.0430 


0.0390 


0.0536 


0.0029 


-0.0340 


-1.318 


0.528 


1.00 


0.9168 


12 .100 


-0.021: 


+0.019: 


0.026: 


0.0007: 


-0.0362: 


-1.404: 


0.50: 



1 Russell's computations were made with a slide rule. Repeating them with five-place 
logarithms, I obtain figures, which sometimes differ slightly from those tabulated. These 
differences, however, are unimportant for they produce no appreciable changes in the final 
elements. 

From the values of h and t% it appears that the observed curve is 
remarkably symmetrical, and that the actual epoch of mideclipse is 
0.0015 days earlier than that assumed by Wendell. The times of 
beginning and ending of the eclipse cannot be read accurately from 
the curve and are marked with colons to denote uncertainty. 

From the values of sin 2 we have now to find k with the 
aid of Table II. From Eq. (13) we have 



<Kfc, «i) = 



sin 2 di - A 
B ' 



hence, if we let A be the value of sin 2 when a x = 0.6 and A — B 
its value when ct x — 0.9, we may find a value of k for every 



ECLIPSING BINARY STARS 



195 



tabulated value of «i by inverse interpolation in Table II. 
Thus, taking A = 0.0369 and B(= sin 2 2 - sin 2 3 ) = 0.0258, 
as given by our curve, we find for ai = 0.0 that 

*(fe,«i) = +4.28: 

and hence, from the first line of Table II, k = 0.56:. Colons 
are here used because the values of k are less accurate when 
the tabular differences of f(k, oi) are small. 

The values of k are seen to be fairly accordant except for 
those corresponding to values a x near 0.6. Inspection of Table II 
"shows that this discrepancy may be almost removed by increas- 
ing all the values of + by 0.024- which may be done by dimin- 
ishing A by 0.0245. Our new value of A is therefore 0.0363." 
The new set of fc's are found to be discordant for values of «i near 
0.9; "but by diminishing B by 2.5 per cent [giving B = 0.0252] 
and hence increasing all the computed values of * in the corre- 
sponding ratio, we obtain a third approximation of a very satis- 
factory character." The general mean is now k = 0.528. 

With these final constants, A = 0.0363, B = 0.0252, k = 0.528, 
we may compute a theoretical light curve and also the elements 
of the system from Eq. (14). Table c gives the second and 
third approximations to the value of k and the data for the 
final light curve. 



Table c 



2d Approx. 



0.0 

0.1 

0.2 

0.3 

0.4 

0.5 

0.6 

0.7 

0.8 

0.9 

0.95 

0.98 

0.99 

1.00 



+4.30: 

2.665 

1.932 

1.314 

0.826 

0.385 

+0.024 

-0.352 

-0.670 

-0.976 

-1.131 

-1.253 

-1.294 

-1.38: 



0.56: 
0.512 
0.517 
0.500 
0.503 
0.47: 

0.48: 
0.40: 

0.714 
0.610 
0.597 
0.55: 



3d Approx. 



+ 4.40: 

2.73 

1.974 

1.344 

0.845 

0.394 

+0.025 

-0.360 

-0.685 

-1.000 

-1.157 

-1.282 

-1.324 

-1.412 



0.58: 
0.534 
0.538 
0.527 
0.532 
0.51: 

0.54: 
0.51: 

0.564 
0.507 
0.512 
0.48: 



+4.100 

2.713 

1.949 

1.348 

0.843 

+0.400 

0.000 

-0.358 

— 0.689 

-1.000 

-1.162 

-1.276 

-1.318 

-1.389 



Final light curve 



^(0.528, ai)\ B+ 



0.1032 

0.0683 

0.0491 

0.0338 

0.0212 

0.0101 

0.0000 

-0.0090 

-0.0173 

-0.0252 

-0.0293 

-0.0322 

-0.0332 

-0.0350 



sin" 



0.1395 
0.1046 
0.0854 
0.0701 
0.0575 
0.0464 
0.0363 
0.0273 
0.0190 
0.0111 
0.0070 
0.0041 
0.0031 
0.0013 



sin $ 


( 


0.373 


0.292 


0.324 


0.252 


0.292 


0.227 


0.265 


0.205 


0.240 


0.185 


0.215 


0.166 


0.191 


0.146 


0.165 


0.127 


0.138 


0.106 


0.105 


0.081 


0.084 


0.064 


0.064 


0.049 


0.056 


0.043 


0.086 


0.028 



196 THE BINARY STARS 

Plotting the magnitudes computed in Table a against the epochs 
—0.0015 ± t, we obtain the computed light curve. The residuals 
(O — C) are given in the last column of Table a. Their average value, 
regardless of sign, is 0?020. 

From Table Ha we find for k = 0.528, $ x (ib) = 0.382, 

<h(k) = 0.507; 
whence 

cot 2 % = -^jjT - A = 0.0133, cot i = 0.115, i = 83° 25' 

n* cosec 2 % = -r-jyr = 0.0660, ri 2 = 0.0652, n = 0.256 
<Pi(k) 

and finally 

r 2 = kri = 0.135. 

In other words, we have, taking the radius of the orbit as unity, 

Radius of larger star . 256 

Radius of smaller star . 135 

Inclination of orbit plane 83° 25' 

Least apparent distance of centers 0.114 

Light of larger star . 0832 

Light of smaller star . 9168 

Period of revolution 4 . 8061 days 

At the middle of eclipse, the larger star overlaps the other by only 
0.007 of the radius of the orbit, or about one-twentieth of the radius 
of the smaller body, so that the eclipse is very nearly grazing. The 
smaller star gives off eleven times as much light as the other, and exceeds 
it fortyfold in surface brightness. 

The loss of light at secondary minimum should be k times the light 
of the fainter star, or 0.023 of that of the system. The corresponding 
change in stellar magnitude is 0.027, which could only be detected by 
refined observations. 

For disks darkened toward the limb the solution proceeds 
along essentially similar lines. The function / must be deter- 
mined by numerical integration. The principal difference is 
that an annular eclipse is not flat-bottomed, since more light is 
cut off when the companion obscures the bright center than 
when it obscures a region near the limb. (An actual case of this 
sort has been reported by McDiarmid, TX Cassiopeiae.*) 

* Contr. Princeton Univ. Obs. No. 7, 1924. 



ECLIPSING BINARY STARS 197 

The changes of light at the beginning and end of eclipse are slower, 
so that the times of first and last contact, calculated from the 
deeper parts of the eclipse tend to be farther apart, and the 
computed diameter of the brighter star, at least, larger than for 
the "uniform" solution. 

Tables for disks in which the brightness falls off to zero at the 
limb have been computed by Russell and Shapley.* 

Orbital eccentricity produces no perceptible asymmetry in 
the form of the light curve for a single minimum but displaces 
the secondary minimum from the half-way point between the 



O 

Sun 



C^^_ 




€> 



Fig 13 —The system of W Delphini. Two relative orbits of the bright star 
are shown, the upper one representing the elements as given in the accompanying 
solution, the lower, Shapley's, on the assumption that the stars are darkened to 
zero at the limb. The diameters of the disks of stars and Sun are drawn on the 
same scale. The three bodies are of equal mass, but the stars are less dense than 
the Sun. {From Shapley's artide in Povular Astronomy, 20, 572, 1912.) 

primary minima, and may change the durations and depths of 
both. It is even theoretically possible for the eclipse to fail 
altogether near apastron. The component e cos w which dis- 
places the epoch of secondary, can often be very accurately 
determined from the observations; the component e sin «, which 
alters the length of minimum, is very hard to find with accuracy. 
In one noteworthy case, Y Cygni, to which reference has 
been made on an earlier pagef a rotation of the line of apsides 
detected photometrically has been confirmed by spectrographs 
observations. 

Ellipticity of the components, caused by their mutual attrac- 
tion, has been found in practically all cases where pairs of small 
separation have been well observed. It causes the light curve 

* Ap. Jour. 36, 239, 385, 1912. 
t See p. 182. 



198 THE BINARY STARS 

to be bowed up between the minima, with maxima halfway 
between them. Its amount increases as the relative distance 
separating the two stars diminishes, in close agreement with 
Darwin's theoretical calculation for masses of homogeneous 
fluid. The computed ellipticity of figure corresponding to a 
given light curve depends, however, on the assumed degree of 
darkening at the limb, so that it is the relative, rather than the 
absolute, values which are of importance. 

For certain systems whose components are almost in contact 
(VW Cephei, etc.) the greater part of the variation is due to 
ellipticity of figure. 

The "reflection" effect* arises from the heating of the com- 
panion by the radiation of the primary, so that the side turned 
toward the latter is brighter than the opposite side. In con- 
sequence, the light curve is higher just outside the secondary 
minimum than outside the primary. This effect is small com- 
pared with the total light but may amount to a very large part 
of the light of the secondary. In Y Camelopardalis, indeed, the 
side of the secondary which is turned away from the primary 
appears to be almost completely dark. 

The information obtained from eclipsing variables is of far- 
reaching value. When supported by spectrographic observa- 
tions, it provides the most complete knowledge of a stellar 
system that we can hope for at present — masses, linear diameters, 
densities, surface brightness (when the parallax is known, as for 
the distant companion of Castor), and even something regard- 
ing the law of internal density from the motion of periastron 
(Y Cygni). Even without this potent aid we may obtain 
data otherwise inaccessible regarding the relation of surface 
brightness to spectral type and color, and especially concerning 
stellar densities. 

The equations for computing the latter are simple. Let the 
total mass of the system be m, that of the larger star my, and 
that of the smaller m(l — y). If a is the semimajor axis of the 
orbit, we shall have a = Km V3 -P % , where £ is a constant 
depending on the units of measurement. If we choose the 
Sun's mass, the Sun's radius, and the day as units, then, for 
the Earth's orbital motion, a = 214.9, P = 365.24, whence 
K = 4.206. 

* First detected by Dugan in the light curve of RT Persei and by Steb- 
bins in that of Algol. 



ECLIPSING BINARY STARS 199 

In determining the elements of the system we have taken a 
as our unit of length. The actual radius of the larger star is 
therefore ari, and its volume, in terms of the volume of the 
Sun, if 3 mP 2 ri 3 , or 74.4raPVi 3 . Hence its density is 

= 0.01344?/ 
and similarly that of the smaller star, 

(i -y) 



p 2 = 0.01344- 



PW 



If the magnitudes and spectral classes are known, the ratio 
y/(l — y) can be estimated closely with the aid of the mass- 
luminosity law. This procedure will give close approximations 
to the true densities, particularly for the brighter and more 
massive components. 

Russell summarizes the conclusions he has drawn from his 
extensive statistical investigations as follows: 

The great majority of them (the brighter components) belong 
to the main sequence, and are of classes B8 to A5. These 
are strikingly similar in density, clustering closely about a mean 
value 0.32 times that of the Sun. (This is the geometric mean 
which in a case like this is more representative than the arith- 
metic.) Every spectral type is, however, represented, the whole 
main sequence, from the M-dwarf YY Geminorum to the super- 
giant O-type stars Y Cygni* and H.D. 1337, f Giants are fewer, 
but cover a wide range of spectral type and density. Only the 
white dwarfs are absent. (It is worth notice in passing that a 
pair of white dwarfs revolving nearly in contact might have a 
period of only two or three minutes !) 

The faint components of eclipsing systems are usually larger 
than their primaries. This is a conspicuous example of observa- 
tional selection — small companions producing only shallow 
eclipses which are unlikely to be discovered. The stars thus 
selected appear to belong to a rather unusual type, intermediate 
between ordinary giants and dwarfs, about which we would 
otherwise know next to nothing. 

In Table 1, I have listed the dimensions, masses and densities 
of 22 systems for which complete spectrographic and photometric 

* Princeton Contrib. No. 12, 1931. 
t Publ. Dom. Ap. Obs. 3, 275, 1926. 



200 



THE BINARY STARS 







._ COIN <N 2 H H N C* <-< Ej 










Jg OJ M O) OS H O O) O! OS 0> 0> O 










^ co h ~ i>~ eo ^ t- oo- -T ^ 22^" S S 




■4^ 


- H ^ if) x H lO »i ~ CO •* ^ i S O ™ °° mm 
N « * ° ri \-H 2 .' H rt M n ffiffl ^N 2 © 2 M S 

--S .a • S • ^ • • -o5s» . o> "* co ■*• o> ^ 

2 «H 5 ^ «^ « ^ "<j ^. ^. g rf 2 ^ So b,^t R.o ^ 


* 

m 

S 


O 


<5 




H 




OQ 




in 




- £ g °l s £ S ^ -g S 3 if a. £ £ E A Z 3 . 3*2 


as 

m 
s 

flu 

M 




(SmPhPhPhcbOm^^PhPhPh^cqcoI-h^^Ii-s^i-sh, 




^oaeoO(MOOOSt^C<IOC^O«OeO<-HO'-HOC<IOOQ>Cp 

iNoooioiNdddooooccowN'ONfioNHNN 

m N i— 1 >- 1 TH I-H i— 1 i-H i— 1 T-t I— 1 1-1 l-l 




. +3 

OQ 0J 


O 

OQ 


a ° 


o 

1— 1 


^ 


§S (O » O 2?5«NNtoO2HH 2! 000^HN 
^ H OOT-H<NOOO^^H'-iOOO^COt*eOOOaiOOO 


H 




ii esocJooodooodoooooooO'-iOT-i 


CQ 

S5 
w 

n 


a, 


— ii — 

089eoS8SlOH««HNHIONHjOH 

50r-H?HrHOO'-l'-H'-'«0'- | ' _lrHC< ' OOWWT ^ QOOrP 






OOOOOOOOOOOOOOOOOOONO'-h 




t» CO CO ?£> *tf hO)h «005t»C3 h 2 -. fc 


Vr> 


OO^^UJtobMHMNtoOOCOWN^WNTtiOiiO 


of 
H 

OS 

as 


£ 


, ~'e005t^T)ir-<(N(N»d»0 - *'-i'-'C<I<NO'-HO'-<'-HO'-HO 




II W H H 


-o 


" QO N ID t« ft tO O CO Q ft ^ "O *Q °? _ °2 
(Offll»iO'ft->0<0<0''5 1 '5HfflNONOHHOHO 


s 


5 




eo i— i >— • 


of 
O 

M 

OS 




« n th to is <o eo io « h oo ec a * PO — 'S 


t* 


H iON^N^>«'*M'*MdNiONHHHMiOOiOO 


-o 


to 10 o«M» woo '-i^SS^E^^SP^S? 

n OO : *NN$S(N*lOINOlO'*00'OM®NtONON 


a 


V. 




r- 1 




<N 


On 

CO 


OHiNMwwMMiooOQOgjaqoiopipoo'OH 


s 




00 •>* ■* 


►J 

9 


eC 


NWOMiOiO»iO»0000HM©00»^2OWt-H 
-jS^OO^OTtlWCOtOTiiWWOJt-OCfflWWO* 


H 


eiJHMciwNNHHHTiii-iMWHNOM'^O'OO 




u 

03 

as 


- Ill ^ If i^plll^^l|l| J i 










HNM^WfONOOOSOHNCO^jOCNOOWg^gj 



ECLIPSING BINARY STARS 201 

data are available, taking the figures from the authorities cited. 
The stars are arranged in the order of spectral class of the 
primary, and the various columns give, in addition to the star's 
name, spectral class and approximate revolution period, the 
radii of the two components (6, bright, /, faint), their masses, 
and their densities, all in terms of the Sun as unit, and the 
distance between their centers in millions of kilometers. 

There are many additional systems for which similar data 
have been or might be computed with the aid of the mass lumi- 
nosity relation, on the basis of the light curves supplemented, in 
some instances, by spectrographic orbits of the brighter com- 
ponent. In fact, in his recent memoir Die Bedeckungsverdnder- 
lichen," Gaposchkin gives such data for 218 of the 349 eclipsing 
variables (many of them as faint as the tenth magnitude) he 
catalogues. In many cases his figures are but rough approxima- 
tions, computed, without even the aid of a light curve, from the 
observed depths of the minima at the two eclipses, and the 
observed durations of the eclipses. For 54 systems he gives 
more precise values for some or all of these quantities. 

Utilizing all of the available data, Gaposchkin discusses not 
only the complicated relations between mass, density, lumi- 
nosity, and spectral type in the eclipsing variables but also the 
general characteristics of their orbits and the question of their 
galactic concentration. He finds, among other results, that 
there is a definite concentration toward the plane of the Milky 
Way, most markedly in the case of the pairs of spectral classes 
O and B, and that the eccentricity of the orbit increases both 
with increasing period and with the spectral class as we pass 
from A toward M. 

References 

Pickering, E. C. : Dimensions of the Fixed Stars with Special Reference to 

Binaries and Variables of the Algol Type, Proc. Amer. Acad. Arts and 

Sciences 16, 1, 1881. 
Roberts, A. W. : On the Relation Existing between the Light Changes and 

the Orbital Elements of a Close Binary System, Mon. Not. R.A.S. 

63, 527, 1903. 
: On a Method of Determining the Absolute Dimensions of an Algol 

Variable Star, Mon. Not. R.A.S. 66, 123, 1906. 
Russell, H. N. : On the Determination of the Orbital Elements of Eclipsing 

Variable Stars, Ap. Jour. 35, 315, 1911; 36, 54, 1912. 
Russell, H. N., and H. Shapley: On Darkening at the Limb in Eclipsing 

Variables, Ap. Jour. 36, 239, 385, 1912. 



202 THE BINARY STARS 

Shapley, H. : The Orbits of Eighty-seven Eclipsing Binaries — A Summary, 
Ap. Jour. 38, 158, 1913. 

: A Study of Eclipsing Binaries, Contrib. Princeton Univ. Obs. 3, 1915. 

Vogt, H.: Zur Theorie der Algolveranderlichen, Veroff. Sternwarte Heidel- 
berg 7, 183, 1919. 
Fetlaar, J.: A Contribution to the Theory of Eclipsing Binaries. 

Recherches Astron. de VObs. d' Utrecht 9, Pt. 1, 1923. 
Sitterly, B. W.: A Graphical Method for Obtaining the Elements of 
Eclipsing Variables, Pop. Astron. 32, 231, 1924; Contrib. Princeton Univ. 
Obs., #11, Pt. I, 1930. 
Scharbe, S. : Bestimmung der Kreisbahnen der Veranderlichen vom Algol- 
typus nach der Helligkeitskurve, BuU. Obs. Central de Russie 10, No. 94, 
1925. 
Gaposchkin, S.: Die Bedeckungsveranderlichen, Veroff. Univ. Stern- 

wartezn Berlin-Babelsberg 9, No. 5, 1932. 
W. Kbat: Some Remarks on the Determination of the Orbital Elements of 
Eclipsing Variable Stars, Veranderliche Sterne, Forschungs- und 
Informations Bulletin 4, 97, 1933. Nishni-Novgorod. The note 
presents a generalization of Russell's method extending it to apply 
also to systems of great eccentricity. The formulas for finding e and a 
from the light curve are free from the assumption that i = 90°. 
Walter, Kurt: Die Bewegungsverhaltnisse in sehr engen Doppelstern- 
systemen, Schriften der Konigsberger Gelehrten Gesellschaft 10, Heft 4, 
1933; or Veroff. Univ. Sternwarte Konigsberg Heft 3. 
Many papers dealing with the orbits of particular systems will be found 
in Bulletins of the Laws Observatory, the Publications of the Dominion 
Astrophysical Observatory, the Contributions from the Princeton University 
Observatory, the Astrophysical Journal, the Astronomische Nachrichten, the 
Monthly Notices of the Royal Astronomical Society, and other journals and 
observatory publications. 

The sections on the Algol variables in such works as Die veranderlichen 
Sterne, by J. G. Hagen, S. J., and J. Stein, (1913-1924) and Geschichte und 
Literatur der veranderlichen Sterne, by G. Muller and E. Hartwig (1918) 
may also be consulted. 

R. Prager publishes annually (Kleine Veroff. der Univ. Sternwarte zu 
Berlin-Babelsberg) a Katalog und Ephemeriden Veranderlicher Sterne, which 
includes Algol variables, and T. Banachiewicz, in the Supplements Inter- 
nationale of the Krakow Observatory Publications, a list of the eclipsing 
variables with ephemerides for those of established orbits. In the issues 
for 1933, Prager gives elements for 582 eclipsing variables, and Banachiewicz, 
light elements and ephemerides for 300, and a list of 722 others, some 
which may prove not to be of the Algol type. 



CHAPTER VIII 
THE KNOWN ORBITS OF THE BINARY STARS 

Several hundred orbits of visual binary stars and of stars with 
periodic variable radial velocity have been computed by the 
methods presented in the preceding chapters. Every computa- 
tion was undertaken with the immediate object in view of 
representing the observed motion and of predicting the future 
motion in the particular system on the assumption that the 
bodies are moving in obedience to the law of gravitation. It 
is satisfactory to find that while the computed orbits exhibit 
the utmost diversity in form and in dimensions, we have found 
no reason to question the validity of that assumption. 

Back of this immediate objective, long since attained, lay the 
broader motive of providing additional data for the study of 
the greater questions of the origin and evolution of the binary 
star systems and of their relation to single star systems. In 
the present chapter we shall examine the computed orbit ele- 
ments, first with respect to correlations that may exist between 
them and then more particularly for the information we may 
derive from them as to stellar masses and densities. 

Not all the orbits which have been computed can be used in 
such studies. The observed arcs upon which many of the 
orbits of the visual binary stars rest are so short that a great 
variety of apparent ellipses may be drawn that will represent 
the data within the error of measure. The well-known binary 
a Coronas Borealis offers a striking example: using practically 
the same data, Lewis found the period to be 340 years, while 
Doberck gave 1,679 years. Even with longer arcs we have, in a 
number of cases, two or more radically different orbits. Reject- 
ing all of these, as well as a few that rest upon assumptions 
which seemed plausible but which have not been supported by 
later observations, we still find in those retained a wide range 
in reliability. The majority are fairly good, some, for all prac- 
tical purposes, are definitive, a few others only slightly better 
than some of those that have been rejected. 

203 



204 THE BINARY STARS 

In Table I in the Appendix, I have listed the 116 pairs retained 
(including Cc** of £ Cancri and Aa of £ Ursae Majoris) giving 
in general the latest* set of elements when two or more have been 
computed for the same pair. The columns of the table give, in 
order, the name of the pair, its position for 1900.0, and the 
magnitudes and spectral class f (or classes), the orbit elements, 
and the authority. 

Similar tables of the best available orbits have been published 
from time to time, the most recent one being the one by W. H. van 
den Bosf). Luplau-Janssen and his colleagues at the Urania 
Sternwarte, Copenhagen, have also published a catalogue! 
containing every orbit published up to 1926. For 21 systems, 
10 or more orbits are listed by them, 70 Ophiuchi leading, with 
33 different sets of elements. 

The orbits of spectroscopic binary stars, based upon measures 
of radial velocity, which in every case cover at least one complete 
revolution period and as a rule a considerable number of revolu- 
tions, are, with few exceptions, more accurate than the orbits of 
visual binaries. But not every star whose observed radial velocity 
varies periodically is a binary system. The observed variation 
may be the result of motion of a periodic character in the atmos- 
phere of a single star. This is apparently true of all those which 
also show variation in brightness of the Cepheid type, and also 
in general, of those in which the period of variation is but 
a fraction of a day, unless they are eclipsing variables like 
W Ursae Majoris. 

Otto Struvelf has given special attention to stars of this class 
and has developed a criterion based upon the relation between K, 
the semiamplitude of the velocity variation, and P, the period. 

When mean values are used for the quantities involved, 
including the masses, this relation || may be written 

K = CP~ H (1) 

It is simply Kepler's harmonic law in different form, and shows 

** = t bnc Cc, the Spectroscopic Binary. 

* The tabulation, however, does not. include orbits published after 
September, 1934, unless they were available to me in advance of publication. 

t From the Henry Draper Catalogue, if the star is listed there. 

X B.A.N. 3, 149, 1926. 

§ Erganzungshefte zu den A.N. 5, Nr. 5, 1926. 

If Ap. Jour. 60, 167, 1924. 

|| The equation from which Struve derives this relationship is given in the 
section on the Masses of the Binary Stars, on p. 218(4). 



THE KNOWN ORBITS OF THE BINARY STARS 205 

that the semiamplitude should vary inversely as the cube of the 
period. Struve finds a very satisfactory agreement between 
the observed and the computed values for K for stars of all 
periods with the exception of those of two classes : the Cepheid 
variables and the stars, not eclipsing binaries, which show 
variable radial velocity with periods of but a small fraction of a 
day. The mean value of K for 10 stars of this latter class is 
very nearly the same as the mean for 15 Cepheid variables 
which he investigated, and in neither class do the values vary in 
accordance with Kepler's law. Excluding the Cepheid variables 
and these short period pseudo-Cepheids, and also a few stars 
of doubtful character, like a Orionis, for which rather uncertain 
orbits of long period have been computed, there remain 326 stars 
which are undoubtedly spectroscopic binary systems. These 
are listed in Table II in the Appendix, which is similar to Table I 
in its arrangement. One orbit only is given for each system, 
but when the spectra of both components are visible on the plates, 
the significant elements for the secondary, if they have been 
computed, are also entered. Eclipsing binaries, in this table, 
are indicated by an asterisk. 

RELATIONS BETWEEN PERIOD AND ECCENTRICITY 

Certain striking characteristics of the orbits in the two tables 
are recognized on the most casual inspection; for example 
the eccentricity of the visual orbits is generally large, that 
of spectroscopic orbits generally small; the periods of the 
former are long — the shortest known so far (if we exclude 
Capella, measured with the interferometer, and pairs like Aa of 
£ Ursae Majoris, in which one component is invisible) being 
4.56 years (531 = jSIOOOAB);* those of the latter generally short, 
ranging, with few exceptions, from a few hours to about 150 days. 

See, Doberck, and others have called attention to the high 
average eccentricity of the visual binary star orbits and to the 
contrast, in this respect, between these orbits and those of the 
planets in the solar system. This appears again in Table I. 
The average value of e for the 116 systems listed is 0.517 and no 

*Kuiper, in his examination of stars of large parallax, has recently 
found that the 9.2 star Wolf 390 (= B.D. -8° 4352, 16 h 50™1, -8° 09', 
class M3e) is a close pair with components of nearly equal magnitude, and 
maximum separation 0"2. His measures indicate that its period of 
revolution is less than two years. 



206 



THE BINARY STARS 



less than seven of the individual values exceed 0.90. The 
average value for the planets, including Pluto (e = 0.249), 
is only 0.08. 

On the other hand, the average eccentricity for the 326 orbits* 
of spectroscopic binaries in Table II is but 0.174. Recalling the 
fact that the periods of the visual binaries are, on the average, 
much longer than those of the spectroscopic, it is natural to try 
to establish a relation between the two elements. Doberck 
long ago presented evidence tending to show that the eccentricity 
of the visual binaries increased with the length of the period; 
Campbell, Schlesinger, Ludendorff, and others have shown that a 
similar relationship exists among the spectroscopic binaries. 

If we order the 116 systems in Table I in the Appendix accord- 
ing to period and eccentricity, we have Table 1. 

Table 1. — Periods and Eccentricities of Visual Binary Orbits 



~-~-~-^_P, years 


0-50 


50-100 


100-150 


150 + 


Sums 


0- 10 

















10- 20 


7 


1 


2 


1 


11 


20- 30 


7 





2 





9 


30- 40 


7 


4 


2 


1 


14 


40- 50 


9 


5 


4 


6 


24 


50- 60 


8 


7 


5 


5 


25 


60- 70 


2 


1 





3 


6 


70- 80 


4 


2 


5 


2 


13 


80- 90 


2 


2 





3 


7 


90-100 


1 


1 


2 


3 


7 


Sums 


47 


23 


22 


24 


116 



Table 2 gives a similar grouping for 324 of the spectroscopic 
binary star orbits listed in Table II, omitting n Orionis and 
e Hydrae because they have been counted among the visual 
binaries. 

The data utilized in these two tables may be summarized as 
in Table 3. 

These three tables, containing 440 pairs, show practically 
the same relations between period and eccentricity as those 
brought out by the corresponding tables in the first edition 
(drawn up in 1917) when only 187 systems were available for 

* Omitting /* Orionis and e Hydrae, which are also visual binaries, the 
mean value is 0.173. 



THE KNOWN ORBITS OF THE BINARY STARS 



207 



study. There is an increase of eccentricity with increasing 
period, on the average, in the orbits of the spectroscopic binaries 
and also in those of the visual pairs, and the average eccentricity 
is decidedly larger in the visual than in the spectroscopic binary 
orbits. Half the values of e, in Table 1, exceed 0.50, and less 



Table 2 



"^-^P, days 


















e~-\^^ 


0-5 


5-10 


10-20 


20-50 


50-100 


100-500 


500 + 


Sums 


0- 10 


100 


26 


19 


8 


4 


10 


4 


171 


10- 20 


14 


10 


5 





2 


10 


6 


47 


20- 30 


2 


9 


10 


4 


3 


2 


5 


35 


30- 40 


1 


3 


7 


3 


2 


2 


5 


23 


40- 50 


1 


2 


1 


6 


1 


3 


3 


17 


50- 60 


1 


2 


4 


3 


1 


5 


1 


17 


60- 70 





1 


2 


2 





2 


2 


9 


70- 80 











3 





1 





4 


80- 90 

















1 





1 


90-100 


























Sums 


119 


53 


48 


29 


13 


36 


26 


324 



Table 3. — The Relation between Period and Eccentricity in Binary 

Systems 



p 


n 


Av.P 


Av.e 


0- 5 d 


119 


2^736 


0.053 


5- 10 d 


53 


7.424 


0.155 


10- 20 d 


48 


13.551 


0.222 


20- 50 d 


29* 


29.779 


0.358 


50-100 d 


13 


69.829 


0.234 


100-500 d 


36 


229.390 


0.279 


500 d + 


26 


, 5? 81 


0.286 


0- 50* 


47 


27.90 


0.444 


50-100* 


23 


72.44 


0.546 


100-150 y 


22 


119.75 


0.530 


150" + 


24 


249.02 


0.615 



♦Eight of these 29 pairs have computed values of e in excess of 0.53; the remaining 
21 values range from 0.0 to 0.49, with a mean of 0.249. 

than one-tenth of all are as low as 0.20; in Table 2, on the other 
hand, two-thirds of the values of e range from 0.0 to 0.20, and 
only 31 of the 324 exceed 0.50. 



208 



THE BINARY STARS 



To bring out more clearly the "scatter" in the values of e 
in orbits having approximately the same period, and, simul- 
taneously, to show that the curve of relationship between P and e 
has no break in its continuity from the short-period spectroscopic 
binaries to the longest period visual binaries for which orbits are 
available, we may arrange the data as in Table 4. Here the 
spectroscopic binaries are gathered into 11 groups according 
to period, and the visual binaries into 9 groups, the successive 
columns giving the number of pairs in each group, the mean 
period and the mean eccentricity. If the values of e are plotted 
against the logarithms of P, and the points are connected we 
shall have an irregular broken line, illustrating the scatter in 
the values of e; but there is no indication whatever that we are 
dealing with more than one relationship curve. 



Table 4 



N 


P 


e 


30 


1^260 


0.051 


30 


2.236 


0.038 


30 


3.248 


0.072 


30 


4.276 


0.060 


30 


6.830 


0.125 


30 


9.528 


0.238 


30 


13.376 


0.181 


30 


23.457 


0.356 


30 


76.090 


0.242 


30 


299.701 


0.268 


24 


2457.36 


0.312 


13 


11?69 


0.455 


13 


24.11 


0.489 


13 


36.89 


0.383 


13 


49.33 


0.529 


13 


77.43 


0.519 


13 


98.44 


0.468 


13 


126 . 91 


0.601 


13 


186.18 


0.579 


12 


308.82 


0.627 



The results of Russell's statistical investigations of the wider 
visual binaries for which no orbits are likely to be available for 
many years or even centuries, are of interest in this connection. 



THE KNOWN ORBITS OF THE BINARY STARS 209 

He finds the average eccentricity for 500 pairs, of average period 
roughly estimated at 2,000 years, to be 0.61, and "of nearly 800 
more with average period of perhaps 5,000 years" to be 0.76. 
We may summarize the data most effectively as follows: 



p 


n 


Av.P 


Av.e 


to 100 d 


262 


13 d 5 


0.147 


100 d + 


62 


1,023.0(= 2?8) 


0.282 


Oto lOCF 


70 


42?5 


0.478 


100* + 


46 


187.2 


0.574 




500 


2,000 ± 


0.61 




800 + 


5,000 ± 


0.76 



But if there is a correlation between the two elements, it is 
not a simple one; disturbing factors evidently enter. We note, 
for example, the outstanding high average eccentricity for the 
29 orbits of spectroscopic binaries with periods ranging from 20 to 
50 days.* The value of e exceeds 0.60 in five of these orbits, 
as well as in three of shorter period. Again, we note that the 
eccentricity of the visual binary OS341,f with a period of only 
19.75 years is 0.96, and that in six other visual orbits of period 
under 50 years J the value of e exceeds 0.70. Orbits of long period 
and small eccentricity are also found both among the visual 
and among the spectroscopic binaries. We have, moreover, 
to reckon with the fact that the tabulated orbits may to a certain 
degree be affected by observational and computational selection 
and to that extent fail to be truly representative of binary star 
orbits in general. In the case of the spectroscopic binaries this 
selection factor may not be very important, although it is easy 
to see that systems of long period, and consequently small values 
of K, are more easily overlooked than those of short period, 
and that of the systems actually discovered, the short-period 
ones are more likely to be selected for orbit computation. 

The situation with respect to the visual binaries may be more 
serious, though here it is not so much a matter of discovery as of 
observation. If the eccentricity of the orbit is high, the com- 

* A similar high appeared in the 1917 tabulation which had 13 systems 
with periods between 20 and 50 days. 

t ADS 11060; cf. T Boo, £186, ADS 9343. 

{ In a letter received after this passage was written, Dr. van den Bos gives 
reasons for believing that the visual binary OS 536 may have a period of 
only 27 years and an eccentricity of 0.98! 



210 THE BINARY STARS 

panion will spend most of its time at or near the apastron end 
of the ellipse and the chances are that the pair will be discovered 
when the apparent angular separation is at or near its maximum. 
It may then show little orbital motion for a number of years, 
particularly if the orbit has a high inclination, and be regarded 
by observers as "practically fixed," and therefore be neglected. 

Again, a high percentage of the known closer visual binaries 
have components that are sensibly equal in magnitude. If 
such a pair has an orbit of great eccentricity, it may readily 
happen that observers will miss the times of minimum separation 
and that the computed orbit will be one of low eccentricity and 
of approximately twice the true period. Two systems, 8 Equidei 
and £ Scorpii, in which this mistake was actually made, are listed 
in Table 1. 

Making due allowance, however, for all these and other possible 
selective effects, for the actual "spread" in the values of e in 
the various period categories, and for the irregularities in the 
tabulated progression, it still remains true that there is a general 
tendency toward greater values of the eccentricity as the period 
increases. This is a fact to be taken into account in theories 
of the origin of the binary stars, particularly if we assume that 
the spectroscopic and the visual binaries are objects of the same 
class. 

RELATIONS BETWEEN PERIOD AND SPECTRAL CLASS 

Campbell, in his study of the spectroscopic binary stars, 
found evidence of a relationship between the period and the 
spectral class; taking the spectra in the order B, A, F, G, K, 
and M, the period increases as we pass from B toward M. 
Before analyzing the present data to see whether they support 
this conclusion, it should be said that in combining the various 
subclasses, I have taken Class B to include subclasses O to B5; 
Class A, subclasses B8 to A3; Class F, subclasses A5 to F4; 
Class G, subclasses F5 to GO; Class K, subclasses G5 to 
K2; and Class M, subclasses K5 to M6. This agrees with the 
Harvard system, except in the inclusion of subclass B8 under 
Class A, but differs somewhat from the grouping adopted by 
Campbell.* 

* Campbell also included a number of systems whose periods were known 
to be "long" or "short," though their orbits had not then been computed. 



THE KNOWN ORBITS OF THE BINARY STARS 



211 



Table 5 shows the distribution with respect to period and 
spectral class of 439 pairs* entered in the preceding tables. 

Table 5. — The Relation between Spectral, Class and Period 



"\Spectrum 
















^\^^ 


B 


A 


F 


G 


K 


M 


Sum 


P ^^\ 
















0- 5 d 


41 


42 


16 


18 


1 


1 


119 


5- 10 d 


13 


23 


1 


11 


5 





53 


10- 20 d 


10 


18 


3 


11 


6 





48 


20- 60 d 


6 


8 


6 


4 


6 





29 


SOrlOO" 1 


3 


4 





5 


1 





13 


ioo-soo 4 


7 


3 


3 


6 


14 


3 


36 


500 d + 


4 


2 





5 


14 


1 


26 


0- 50" 





8 


4 


26 


8 


2 


47 


60-100" 





4 


1 


13 


5 





23 


100-160" 


1 


4 


3 


9 


4 


1 


22 


150* + 





6 


5 


7 


6 


1 


2 


Totals 


85 


122 


42 


114 


68 


9 


440 







It will be noted that, while there is a wide range in period in 
pairs of every one of the spectral classes, more than two-thirds 
of the spectroscopic binaries with periods of ten days or less 
belong to spectral classes B or A, that pairs with periods in 
excess of 100 days are most numerous in Class K, whereas Class M 
has, in all, but five representatives. In the visual orbits, on the 
other hand, Class G is best represented, especially in pairs with 
periods not exceeding 100 years. Class M, again, has few 
representatives and but one Class B pair appears. 

To investigate the question of progression in period with 
advancing spectral class, I have computed the average periods 
of pairs of each spectral class, but in doing so have omitted four 
spectroscopic binaries of Class B, two of Class A, and one of 
Class G, because of their abnormally long periods. The results 
are given in Table 6, which records also the average eccentricity 
for each group. If the pairs just referred to were retained, the 
average period for Class B (84 stars) would become 149^9, for 
Class A (100 stars), 36^82, and for Class G (60 stars), 274^20. 
The average eccentricities would be but slightly changed. 

*The spectral class of the primary star, in visual binaries, is the one 
tabulated. The system £ Urs. Maj. Aa is therefore omitted, since the 
primary has already been counted. 



212 



THE BINARY STARS 



Table 6. — The Relation between Spectral Class and Average 
Period and Eccentricity 



Class 


Av.P 


Av. e 


N 


B 


24?61 


0.153 


80 


A 


15.08 


0.162 


98 


F 


25.80 


0.225 


29 


G 


101.20 


0.152 


59 


K 


681 . 12 


0.212 


46 


M 


336.07 


0.096 


5 


B 


104 y 3 


0.314 


1 


A 


120.3 


0.565 


22 


F 


133.4 


0.566 


13 


G 


84.0 


0.518 


54 


K 


97.5 


0.478 


22 


K5 


167.45 


0.556 


2 


Ma, b, 


43.7 


0.296 


2 



It again appears that the spectroscopic binaries of the "early" 
spectral classes have, on the average, short periods, those of the 
"late" classes, long periods; but the progression is irregular, 
the pairs of Class A having the shortest periods, those of Class K, 
the longest. Except for the fact that pairs of Classes G and K 
have, on the average, the shortest periods,* the visual binaries 
give little evidence of correlation between period and spectral 
class. 

It has already been pointed out that spectroscopic systems of 
short period are more readily discovered than those of long 
period, for not only is the amplitude of the velocity curve greater, 
in general, in the former, but the variation in the velocity becomes 
apparent in a much shorter time. We may expect relatively 
more long-period systems in future discoveries among stars of all 
classes of spectra and hence an increase in the average values of 
the periods. It should also be noted that the spectra of stars of 
the later types, in general, show more lines, and these more 
sharply defined, than the spectra of the early-type stars. The 
probable error of measure is therefore less and hence a variable 
radial velocity of small amplitude may be unmistakable in, 
say, a Class G or K star, whereas one of equal amplitude may 
escape* recognition in a star of Class B. This may account, in 
part, for the distribution in spectral class shown by the longer 

* Unless we count the two M type stars m Herculis BC, and Krueger 60. 



THE KNOWN ORBITS OF THE BINARY STARS 213 

period spectroscopic binaries in Table 5, but it obviously does 
not explain the large number of short-period binaries of classes B 
and A. 

The decrease in the average eccentricity of the visual binaries 
with the advance in spectral class from A,F to G,K, is a curious 
feature of Table 6. It is apparently a selective effect, for the 
percentage decrease is definitely smaller than in the corresponding 
table in the first edition which listed 68 pairs. The spectroscopic 
binaries show no such progression. 

Absolute trigonometric parallaxes are available for 89 of the 
visual pairs listed in Table I. I have computed the absolute 
magnitudes for the primary components in these systems, and 
have divided them into two groups at M = 3.0, which is ordi- 
narily taken as the point of division between giant and dwarf 
stars. Omitting the exceptional white dwarf star, 40 Eridani 
BC (M = 10.7), we have the following table: 

Table 7. — The Relations between Absolute Magnitude, Spectkal 
Class, Period, and Eccentricity 



M<L 3.0 


M > 3.0 


Sp 


JV 


M 


P 


e 


n 


M 


P 


e 


B 


1 


-0.4 


104 y 3 


0.31 











A 


19 


+ 1.53 


111.1 


0.58 


1 


+3.2 


139 f3 


0.59 


F 


8 


2.28 


134.3 


0.58 


3 


3.33 


156.3 


0.46 


G 


9 


2.26 


55.3 


0.51 


29 


4.42 


84.6 


0.46 


K 











16 


5.71 


111.8 


0.46 


Ma, b 











2 


11.00 


83.2 


0.30 



The distribution of stars of the different spectral classes in 
the two groups occasions no surprise, but it is of interest to note 
that fully two-fifths of the pairs fall into the group with M ^ 3.0. 
In classes F and G, the average P for pairs in this group is smaller 
and the average e larger than for pairs in the group M > 3.0; 
but these facts, especially in view of the small numbers of pairs 
involved, are probably without significance. 

THE DISTRIBUTION OF THE LONGITUDES OF PERIASTRON 

In 1908 Mr. J. Miller Barr called attention to a singular 
distribution of the values of o>, the longitude of periastron, in 
those spectroscopic binaries whose orbits are elliptic. In the 



214 



THE BINARY STARS 



30 orbits available to him in which e was greater than 0.0, 
26 had values of o> falling between 0° and 180° and only four 
between 180° and 360°. He concluded that the effect was due 
to "some neglected source of systematic error" in the observed 
radial velocities, but both Ludendorff and Schlesinger, examining 
the data, were of opinion that it "was nothing more than a 
somewhat extraordinary coincidence," for it became less marked 
as additional orbits were computed. 

In more recent years this question has been discussed by a 
number of investigators, some offering theoretical explanations 
for the unequal distribution which they regard as real, others 
refuting the explanations advanced. The history of these 
discussions, with full references, is given by O. Struve and 
A. Pogo* in a paper they published in 1929. Their own investiga- 
tion leads them to the conclusion that the observed distribution 
may be real and may arise from conditions in the stellar systems 
that would produce a tendency toward a particular orientation 
of the periastra with respect to the direction to the center 
of the galactic system, but they admit that the evidence is not 
conclusive. 

I have examined the data in Table II with respect to the dis- 
tribution of the values of «, with the result given in Table 8. 



Table 8 



e -— 


0°-90°0 


90°-180?0 


180°-270!0 


270°-360t0 


0-0.10 
0.10-0.20 
0.20-0.50 
0.50-0.88 


47 

8 

28 

10 


27 

13 

14 

6 


22 
10 
21 

7 


24 
15 
13 
10 


Totals 


93 


60 


60 


62 









All orbits in Table II, except those definitely noted as circular 
and two in which o> was set down as variable, are included. It 
appears that the number of values for « in the first quadrant 
exceeds by 50 per cent the number in any of the other three 
quadrants, but that the excess is most marked in the orbits 
for which the value of the eccentricity is not greater than 0.10; 

* "Ober die Ursache der ungleichen Verteilung der Periastronlangen bei 
spektroskopischen Doppelsternen, A.N. 234, 297, 1929. 



THE KNOWN ORBITS OF THE BINARY STARS 215 

that is, in those orbits for which the value of oj is least deter- 
minate. For the orbits with e > .10, the sums, in the four quad- 
rants, are, respectively, 46, 33, 38, 38, and the excess in the 
first quadrant is but 28 per cent. Even this is rather larger 
than would be expected in a purely random distribution of values, 
but it will be well to wait until a much larger number of accurate 
orbits becomes available before accepting it as proof of a real 
inequality in the distribution of the periastra. 

THE ORIENTATION OF THE ORBIT PLANES OF THE VISUAL 

BINARY STARS 

A related problem is that of the orientation of the orbit planes 
of the visual binary stars. A number of investigations have 
been made to ascertain whether these orbit planes exhibit a 
random distribution or whether there is a tendency to parallelism 
to a particular plane as, for example, the central plane of the 
Milky Way. 

Practically, the problem is to determine the distribution of the 
poles of the orbits, and in its solution we encounter the serious 
difficulty that the orbit elements of a binary star do not define 
its plane uniquely unless the inclination is 0° or 90°. It is only 
when the indetermination in the sign of the inclination has been 
removed by spectrographic observations that we can discriminate 
between the true and the "spurious" pole. For this reason, the 
conclusions reached by the earlier investigators are all open to 
question, and it is not surprising that they differ widely. Miss 
Everett,* See,f and Doberckf found the distribution to be a 
random one: Lewis and Turner § concluded that the evidence 
indicated, somewhat doubtfully, a tendency of the poles to 
group themselves along the Milky Way; Bohlin^[ noted a division 
into two groups, one with a concentration of the poles near the 
pole of the ecliptic and the solar apex, the other with a con- 
centration near the pole of the Galaxy. The more recent 
investigations by Kreiken|| and Shajn** also indicate a con- 
centration near the pole of the Galaxy. 

* Alice Everett, Mon. Not. R.A.S. 56, 462, 1896. 

t T. J. J. See, Evolution of the Stellar Systems, 1, 247, 1896. 

t W. Doberck, A.N. 147, 251, 1898; A.N. 179, 299, 1908. 

§ T. Lewis and H. H. Turner, Mon. Not. R.A.S. 67, 498, 1907. 

1f K. Bohlin, A.N. 176, 197, 1907. 

|j E. A. Kreiken, Mon. Not. R.A.S. 88, 101, 1927. 

** G. Shajn, Mon. Not. R.A.S. 86, 643, 1925. 



216 THE BINARY STARS 

All of these studies rest upon the known orbits of visual binary 
stars. Professor J. M. Poor,* on the assumption that parallelism 
of the orbit planes would reveal itself as a variation in correlation 
between position angle and distance of double stars in different 
parts of the sky, based a statistical study on all the data available 
in 1913 and concluded that there is a concentration of poles near 
the vertex of the preferential motion of the stars. 

Quite recently Y. C. Changf and W. S. Finsen $ have investi- 
gated the question using as data only those orbits for which the 
true pole is known. Chang, in 1928, based his study upon 
16 pairs, including Capella, and the two systems 42 Comae 
Berenices and £ Cawcn,§ whose orbit planes are, respectively, 
approximately parallel and perpendicular to the line of sight. 
Finsen, in 1933, found 28 pairs available, including five for which 
the values of cos i lay between 0.95 and 1.00 or between 0.00 
and 0.05. Neither investigator found any striking concentration 
of the poles, and both conclude that the distribution is probably a 
random one. We may accept that as the best answer to the 
question on the basis of existing data. 

THE MASSES OF THE BINARY STARS 

The only direct method we have of determining the mass of a 
celestial body is to measure its effect upon the motion of another 
body. It follows that the binary stars are the only ones whose 
masses we can determine directly. Since a knowledge of stellar 
masses is fundamental in all studies of the dynamics of the 
stellar system, the methods by which we calculate the absolute 
and the relative masses of the components in the visual and 
spectroscopic binaries merit careful attention. 

Unfortunately, the orbit elements alone do not afford all 
the data necessary for the determination of either mass or 
density. The well-known harmonic law 

D 3 :d 3 = P 2 (M + JWi):p 2 (m + mi), 

will give the mass of any system in terms of the Sun's mass 
when the linear dimensions of the system as well as the orbit 
elements are known. But the semimajor axis of the visual 

* J. M. Poor, A J. 28, 145, 1914. 

t Y. C. Chang, A J. 40, 11, 1929. 

% W. S. Finsen, Communicated in manuscript form in August, 1933. 

§ Schnauder's orbit. 



THE KNOWN ORBITS OF THE BINARY STARS 217 

binary stars is known only in terms of seconds of arc, and its 
value, so expressed, must be divided by its parallax to reduce 
it to linear measure;* and we do not know the true semimajor 
axis of the spectroscopic binary orbits at all, but only the 
function a sin i. This, however, is expressed in kilometers. 

The parallax of a number of visual binaries is known with a 
greater or less degree of certainty, and mass values for those 
systems may be computed, using the harmonic law in the form 

a 3 

(m + mi) = ^p 2 (1) 

in which r is the parallax of the system, P the period and a 
the semimajor axis of its orbit, and the units of mass, length, 
and time are, respectively, the Sun's mass, the astronomical 
unit, and the year. 

While we are unable to derive the mass of any given spec- 
troscopic binary until we have a knowledge of the value of i, 
the inclination, we may nevertheless estimate the average mass 
of a number of systems with approximate accuracy, by deter- 
mining the probable average value of i and hence of sin i. The 
formulas required differ for the two cases (1) when both spectra 
have been observed, and (2) when only one spectrum is visible. 
They may be derived from the well-known relation 

, . 4ir 2 (a + «i) 3 , n s 

(w + mi) = *£r pi (2) 

in which t denotes, not the parallax, but the circumference of 
radius unity, k the Gaussian constant (log 8.23558), a and ai, 
the major semiaxis of the orbits of the primary and secondary, 
respectively, and P their revolution period expressed in mean 
solar days. Since we do not know a but only the function a sin i, 
we must multiply both members of (2) by sin 3 i, and since a sin i 
is expressed in kilometers, we must divide its value by that of 
the astronomical unit expressed in kilometers. The numerical 
value of 4ir 2 /k 2 A 3 is approximately f 4/10 20 and we therefore have 

, . n • 3 • 4 (a sin i + a x sin i) 3 , , 

(m + mi)sm 3 % = ^ • ^ 2 (3) 

* This gives the length in astronomical units. The astronomical unit or 
the Earth's mean distance from the Sun is, in round numbers, 149,500,000 
km. 

f The more precise value 3.99455/10 20 is used in obtaining the logarithm 
in Eq. (4). 



218 THE BINARY STARS 

From Eq. (9) of Chap. VI, 

a sin i = [4.13833]#P\/r=T 2 
hence 

(m + m x ) sin 3 i = [3.01642 - 10]^ + K X )*P(1 - « 2 ) ? * (4) 

the numbers in square brackets being logarithms. This equa- 
tion is independent of the parallax, or distance of the system. 

When both spectra have been measured and the correspond- 
ing velocity curves drawn we obtain at once the relative masses 
of the two components, from the relation m:m x = K X :K; 
and we also have the equations 

m sin 3 i = [3.01642 - lORX + K x yK x P(l - e*)*\ 

m x sin 3 * = [3.01642 - 10] (K + K X )*KP(1 - e*) H j W 

from which to compute the masses of the components separately. * 
When only one spectrum is visible we must apply a some- 
what different formula, viz., 

m x 3 ... 4 (a sin i) 3 .„. 

sm l = jm — m — C6) 



(m + wii) 2 10 20 

in which a sin i and m refer to the component whose spectrum 
is given. We may write this in a form similar to Eq. (4) thus: 

mi 3 sin 3 z = [3 Q1642 _ 1Q]K3 p^ _ eV)H (?) 

(m + mi) 2 ' v ' 

In applying Eqs. (4) and (7) it is necessary to assume a value 
for sin 3 i and the question of obtaining such a value has next 
to be considered. "It can be shown for an indefinitely great 
number of binary systems whose orbital planes are distributed 

* It is possible, in the case of certain eclipsing binaries, to determine the 
value of the velocity range (/f 2 ) of the fainter star even when the direct 
measures of its velocity do not in themselves suffice to define the velocity 
curve. It is only necessary to have enough observations to define the slope 
of velocity curve of the secondary relatively to that of the primary. From 
this relation slope and the orbital elements of the primary, the value of K 2 
can be computed, and thus the mass ratio of the two components. Joy, for 
example, employed this method in his work on the orbit of U Sagittae 
(Ap. Jour. 71, 336, 1930). Here the larger (and fainter) star passes nearly 
centrally in front of the smaller primary and the total eclipse of the latter 
lasts about 100 minutes. With the 100-in. reflector enough spectrograms 
were secured in this short time interval to permit the slope of the velocity 
curve to be determined with considerable accuracy. The number of stars 
in which this method may be employed is, however, small. Contrib. Mt. 
Wilson Obs., #401. 

[See UOC, #68, Feb. 26, 1926, p. 354.— J.T.K.] 



THE KNOWN ORBITS OF THE BINARY STARS 219 

at random, that the average inclination would be 57?3, in 
accordance with the formula 

x X 

2 C 2 P 
io = - I I * sin idid4> = 1 
*"Jo Jo 

The average value of sin 3 i, however, would not be sin 3 57?3 
( = 0.65) but approximately 0.59 in accordance with the formula 

X X 

sin 3 i Q = - I I sin 4 idid<f> = Jie* — 0.59" 

Campbell, whom we have just quoted, and Schlesinger, who, 
from a slightly different formula obtains the same value for 
sin 3 to, point out that while this mean value holds for orbits in 
general, it would not be permissible to use it for the spectro- 
scopic binary stars whose orbits have so far been computed. 
For, to quote again from Campbell, "there is the practical con- 
sideration that binary systems whose orbital planes have large 
inclinations are more readily discoverable than those whose 
inclinations are small . . . Under ordinary circumstances, and 
when dealing with a considerable number of orbits, a compromise 
value of sin 3 i = 0.65 might in fairness be adopted." For 
18 systems which he actually considers he adopts the higher 
value 0.75 because six of them are eclipsing binaries, with 
inclinations quite certainly between 60° and 90°. 

Schlesinger, assuming "that the chance of discovery is pro- 
portionate to sin i" obtains sin 3 i = 0.68 for a mean value. 
We may then adopt, for convenience in computation, 

sin 3 i = 0.667 = %, 

since comparatively few eclipsing binaries are among the number 
under discussion. 

Both spectra are visible in 103 of the 321 pairs in Table II 
and for these the computers of the orbits have given the values 
m sin 3 i, mi sin 3 i, or, for some of the eclipsing binaries, the values 
m and w»i. Table 9 lists these pairs with their spectral classes 
and mass values. The most massive pair, by far, is BD + 57°28,* 
Class B5, and the least massive is probably the eclipsing pair 
S Antliae, Class F0, with masses 0.52 O and 0.29 O for the two 
components, respectively. Smaller minimum mass values are 

* [See p. 251; also cf. Krueger 60, p. 242, a Visual Binary; also (46) Dra, 
B4745, Spec. Binary: Lick Obs. Butt. 483, I, #275 (non-eclipsing).^J.T.K.] 



220 



THE BINARY STARS 
Table 9. — Masses: Spectroscopic Binaries 











2 Spectra 


Star 


Spec. 


m sin 3 i 


mi sin 3 t 


m 


mi 


+57?28 


B5 


113.2 


44.9 






* Boss 46 


B0 


17.57 


16.37 






* TVCas 


B9 






1.83 


1.01 


■*• Cas 


A5 


1.35 


1.34 






7 And 


B3 


1.50 


1.10 






Boss 373 


F5 


1.16 


1.06 






k Ari 


A0 


0.14 


0.13 






i Tri, br m 


GO 


1.12 


1.12 






i Tri fr 


F4 


0.91 


0.86 






+59°609 


B5 


18.88 


9.17 






Boss 816 


B8 


2.87 


2.76 






o Per 


Bl 


5.42 


3.79 






A Per 


F5p 


1.01 


0.88 






* +33?785 


B3 


4.86 


4.29 






Boss 1001 


B9 


0.56 


0.55 






+7?676 


B5 


7.0 


3.7 






Boss 1213 


B9 


2.5 


2.2 






* TT Aur 


B5 


6.7 


5.3 






a Aur 


GO 


1.19 


0.94 






2 674A 


F5 


1.40 


1.33 






Boss 1275 


A0 


1.71 


1.50 






* i) Ori 


Bl 


11.2 


10.6 






<t> Ori 


B2 


5.53 


4.19 






Boss 1457 


A0 


0.63 


0.44 






Boss 1464 


B2 


10.3 


3.9 






P Aur 


AOp 






2.38 


2.34 


-3?1413 


B5 


6.2 


4.1 






* WW Aur 


A0 


2.2 


1.9 






+6?1309 


BOp 


75.6 


63.3 






29 CMa 


Oe 


32.2 


24.3 






Boss 1906 


B8 


4.3 


2.3 






Boss 1945 


F5 


1.05 


0.85 






* a Gem C 


Mle 


0.63 


0.57 






+34? 1657 


F0 


1.53 


1.32 






+20?2153 


A0 


1.39 


1.35 






Boss 2484 


A0 


1.48 


1.27 






* S Ant 


F0 


0.52 


0.29 






o Leo 


F5 


1.30 


1.12 






* WUMa 


GO 


0.67 


0.48 






Boss 2830 


F2, A3 


0.28 


0.24 






uUMa 


A0 


3.50 


0.60 






Boss 2987 


A2 


0.12 


0.08 






Boss 3138 


B3 


8.2 


4.4 






fl'Cru 


A5 


0.74 


0.61 






+74?493 


G5 


0.80 


0.70 






Boss 3323 


A5 


4.62 


2.37 






Boss 3354 


A0 


2.47 


2.08 






* RS CVn 


F8 


1.79 


1.66 






fUMa 


A2p 


1.70 


1.62 






a Vir 


B2 


9.6 


5.8 






Boss 3555 


F5 


2.34 


1.92 






Boss 3635 


F5 


1.36 


1.29 






39 Boo ftr 


F5 


1.27 


1.03 







THE KNOWN ORBITS OF THE BINARY STARS 
Table 9. — {Continued) 



221 



Star 


Spec. 


m sin* t 


mi sin' « 


m 


mi 


*UOrB 


B8 






4.27 


1.63 


f CrB br. 


B8 


13.35 


13.06 






ff Sco 


Bl 


13.0 


8.3 






«■ CrB br. 


GO 


0.94 


1.07 






+17?3063 


A0 


2.19 


1.35 






Boss 4247 


F2 


1.11 


0.99 






cHer 


A0 


1.6 


1.0 






*UOph 


B8 






5.31 


4.66 


* u Her 


B3 


7.5 


2.9 






• TXHer 


A5 






2.04 


1.77 


Bom 4423 


F0 


0.88 


0.82 






+ 14?3329 


A3p 


1.83 


1.62 






*Z Her 


F5p 


1.5 


1.3 






Boss 4602 


F5 


0.46 


0.41 






Boss 4622 


F0 


1.04 


1.01 






Boss 4643 


A2 


1.72 


1.18 






* RX Her 


AO 


2.08 


1.85 






+65?1276 


A3 


1.97 


1.87 






+49?2871 


F5, A 


1.48 


1.47 






Boss 4788 


AO 


0.95 


0.90 






-10.4926 


B5 


7.10 


4.43 






+ 16.3758 


F5 


1.26 


1.26 






*RS Vul 


B8 






5.26 


1.64 


USag 


B9 






6.7 


2.0 


+37?3413 


AO 


1.18 


0.84 






*Z Vul 


B3 






5.25 


2.37 


Boss 4947 


AO 


0.91 


0.65 






* 9 Aql 


B3 


5.3 


4.4 






Boas 6026 


F5 


1.46 


1.44 






+35?3970 


BO 


13.85 


12.90 






Aql 


AO 


0.52 


0.38 






Boss 6173 


A2 


2.27 


2.06 






+45?3139 


Bl 


2.90 


2.35 






• YCyg 


B2 






17.4 


17.6 


Boss 6375 


B3 


1.79 


1.67 






+32-4134 


AO 


1.87 


1.08 






+27?4107 


FO 


0.97 


0.77 






Boss 5575 


A3 


1.62 


1.54 






Boss 5579 


AO 


0.96 


0.95 






Boss 5591 


AS 


1.19 


1.17 






Boss 5629 


B3 


20.8 


13.6 






* RT Lac 


G5 


1.9 


1.0 






Boss 6683 


F5 


0.65 


0.61 






Boss 5764 


B5 


0.87 


0.71 






Boss 5834 


B3 


6.01 


3.87 






Boss 5846 


GO 


1.47 


1.38 






* +64°1717 


B3 


11.4 


9.8 






+58?2546 


B3 


4.8 


2.9 






Boss 6142 


BO 


18.5 


12.7 






Boss 6148 


F5 


1.70 


1.67 






Boss 4745, 46(c)Dn 


AO 


0.12 


0.10 


(perhaps the least massive 
star known) 



The asterisk preceding star names in the first column indicates that the star is an eclipsing 
binary. 



222 



THE BINARY STARS 



listed, it is true, but the factor sin 3 i may, in these cases, also 
be far below the average value. 

It will be noted that two pairs, one in Class B0-B2 (+57°28), 
the other in Class B3-B5 ( + 6° 1309), are extraordinarily massive 
and cannot, therefore, be used in deriving mean mass values for 
stars of those classes. Omitting them, we have the mean values 
given in Table 10. 

Table 10. — Mean Mass Values for Spectroscopic Binary Stars 







Noneclipsing 






Eclipsing 




Class 


















N 

1 


m sin 3 i 


wi sin 3 i 


wi/m 


N 


m sin 3 i 


mi sin 3 i 


m,\/m 





32.2 


24.3 


0.75 










B0-B2 


10 


10.79 


8.09 


0.75 


1 


17.4 


17.6 


1.01 


B3-B5 


16 


7.44 


4.77 


0.64 


1 


5.25 


2.37 


0.45 


B8-A3 


27 


2.08 


1.61 


0.77 


8 


3.75 


2.12 


0.57 


A5-F4 


11 


1.33 


1.04 


0.78 


2 


1.28 


1.03 


0.80 


F5-G2 


18 


1.26 


1.16 


0.92 


3 


1.24 


1.15 


0.93 


G5 


1 


0.80 


0.70 


0.88 


1 


1.9 


1.0 


0.53 


Mt 










1 


0.63 


0.57 


0.93 



In compiling this Table, I have taken the mass values for all 
stars noted as eclipsing binaries to be the true masses, although 
for a number of them the figures in Table 9 are entered as mini- 
mum values. 

Inspection of that table shows that in only two systems 
(Y Cygni, B2 and a Coronae Borealis,btr, GO) is mi slightly more 
massive than m, and that in only 17 others is there practical 
equality between the masses of the two components. The 
general rule is that the secondary is definitely the less massive 
star. We shall see that this holds true, too, for the visual binaries 
for which the mass ratio has been computed. 

From Tables 9 and 10 it is clear that binaries of classes O to B5 
are decidedly more massive than those of later classes and that 
there is a fairly definite progression in the average mass values 
as we pass from O to G, though too much stress must not be laid 
upon the particular figures in Table 10, since there is a large 
range in the individual values for every class. 

Conclusions, moreover, that are drawn from systems in which 
the spectra of both components are recorded cannot, legitimately, 
be extended to all spectroscopic binary systems, for the double- 



THE KNOWN ORBITS OF THE BINARY STARS 



223 



line systems are selected, in the sense that it is only in systems 
with relatively large values of K that the spectrum of the second 
component is visible. The sum (K + Ki) enters by its cube in 
Eq. (4) and the mass, therefore, in general increases very rapidly 
with K. 

The value of the function mj sin 3 i/(m + wii) 2 is frequently 
omitted by the computer of orbits for it gives very little definite 
information. Equation (7) affords a ready means of computing 
the function for any system, but I have not considered it neces- 
sary to carry out the computation, for a glance at the numbers 
recorded in Table II* shows at once that no conclusions could 
be based upon any means that might be taken. There is nothing 
novel in these conclusions; they simply confirm, on the basis 
of more extensive data, conclusions reached by several earlier 
investigators. As early as 1911, for example, Ludendorff 
found, from the systems then available for study, that those of 
Class B were, on the average, about three times as massive as 
those of classes A to K. 

The last two columns of Table I* give the parallaxes and 
masses of 83 visual binary star systems. The parallaxes for the 
brighter stars were taken, with but one or two exceptions, from 
Schlesinger's Catalogue of Bright Stars; those for the stars too 
faint to be listed in that catalogue, from parallax data kindly 
sent to me from the Yale University Observatory. It is to be 
noted that the latter are not Schlesinger's values but my own 
deductions from the data. Taking them all at face value we 
have the following summary: 



Table 11 






Class 


JV 


(m + mi) 


M 


B 


1 
21 
11 
37 
16 

2 


10.65 
4.03 
2.59 
2.43 
2.14 
0.62 


-0.10 


B8 to A3 


1.04 


A5to F3 


2.18 


F5to G2 


3.25 


G5 to K2 


5.25 


Ma, b 


10.70 







The final column gives the mean absolute magnitudes of the stars. 

The means for classes A, F, and G agree well with those for the 
corresponding classes of spectroscopic binaries in Table 10. The 
* In the Appendix. 



224 THE BINARY STARS 

numbers for classes B, K, and M are too small to give the means 
any weight. 

Several of the individual mass values in Table I relate to triple 
or quadruple systems. It is known that one component in each 
of the systems Ho212 (13 Ceti), A2715 (/* Ononis), k Pegasi, 
and 0282, and both components in the systems Castor (AB) 
and £ Ursae Majoris (AB) are spectroscopic binaries. Further, 
invisible companions have been suspected in the system 70 
Ophiuchi and f Cancri (AB) and a fainter companion to the 
companion of Sirius has been reported by several observers. 
It is not at all improbable that other systems may be found to 
have similar additional components when later measures give 
us more accurate orbits. In some instances, the mass of the 
invisible companion is known to be very small, but even so, 
when the data become more extensive and reliable, account must 
be taken of all of these extra bodies in any discussion of the 
masses of the visual binary stars. At present it will suffice to 
call attention to their existence and to the fact that allowance 
for them would modify slightly the figures in Table 11. The 
uncertainties still attaching to the values of the parallax are, 
however, far more important, for many of the parallaxes are 
small and changes in their value, within the probable error 
of measure, will make large changes in the computed masses 
since the parallax enters the formula by its cube. Changes 
in the orbit elements P and a will have far less effect for, to a 
large degree, they will offset each other, since in general they 
vary in the same sense. 

It may be remarked that the comparatively small range in 
mass in the visual binary systems might have been predicted 
from the fact that, in general, long-period orbits have the larger 
values of a, short-period orbits, the smaller ones. A large 
percentage of the known visual orbits are comparable in size 
to the orbits of the major planets in the solar system. 

The parallaxes utilized in Table 11 do not depend upon the 
fact that the stars to which they relate are binary systems, but 
it is to be noted that when, in a visual binary we have not only 
the orbit elements but also spectrographic measures of the 
relative radial velocities of the two components, we have the 
data for an independent determination of the parallax, as See 
pointed out many years ago. Hussey used this method to derive 
the parallax of S Equulei, and Wright, to derive that of a Centauri, 



THE KNOWN ORBITS OF THE BINARY STARS 225 

their values agreeing closely with those given in the table. The 
computation is readily made by means of the following formulas, 
adapted by Wright from the work of Lehmann-Filh6s:* Let 
R = the astronomical unit, expressed in kilometers. 
a = the semi-major axis of the binary, expressed in kilo- 
meters, and a", the same element expressed in seconds 
of arc. 
n = the mean angular motion of the star, in the visual orbit, 
in circular measure per second of time. 
AV = the observed difference in the radial velocity of the two 

components. 
Then 

2tt 

n ~ 86400 X 365.26 X P 

Ayvr^? \ (8) 

n sin i[e cos w + cos (v + «)J 

a 

The micrometric measures connecting the two components of 
a visual double star afford data for the computation of the 
relative orbit only, and give us no direct information about the 
position of the center of gravity of the system or the mass ratio 
of the two components. This information can be obtained only 
from measures connecting one, or both, of the components with 
independent stars. Such measures, covering a sufficient time 
interval, afford the data for the computation of the absolute orbit 
of the component concerned, and thus, by comparison, of the 
mass ratio. 

The classic illustration is the star Sirius. Bessel, in his dis- 
cussion of the proper motion of the bright star, based upon the 
meridian-circle observations, noted as early as 1834 that it was 
not moving uniformly along a straight line but was describing 
a wavy line across the sky. He inferred the existence of an 
unseen companion, the two components revolving about their 
common center of gravity in a period of fifty years, f The 
faint companion discovered by Alvan G. Clark in 1862 fully 
confirmed this prediction, and the combination of the micrometer, 

* I have made slight changes in Wright's notation as given in L.O.B. 1, 

4, 1904. 

f For a more detailed statement, see p. 237. 



226 THE BINARY STARS 

meridian-circle, and parallax measures permits the computation 
of the mass of each component and also of the linear dimensions 
of the orbits. 

When a series of micrometric or photographic measures con- 
necting the components of a binary with an independent star is 
available, the relative masses can be determined in a very simple 
manner.* 

Let AB be the binary system, C an independent star, and 
let p, 6 and p', 0', respectively, be the distance and position 
angle of C referred to A and of B referred to A. Then the 
apparent rectangular coordinates of C and B referred to axes 
drawn from A as origin in position angles O and (90° + O ) 
will be 



x = p cos (0 — do) 


x' = p' cos (0' — do) 


y = p sin (0 — Bo) 


y' = p' sin (6'- do) 



Now if we let K equal the mass ratio B/(A + B), the coordinates 
of the center of gravity of AB will be Kx r , Ky', and since the 
motion of C with respect to this point must be uniform, we have 

x = a + b{t - to) + Kx') y = a' + V(t - t ) + Ky', (9) 

to being any convenient epoch. 

Each set of simultaneous observations of AB and AC fur- 
nishes an equation of condition in x and one in y for the deter- 
mination of the five constants a, 6, a', &', K. No knowledge 
of the period or other elements of the binary system is involved, 
the accuracy of the determination of K depending entirely 
upon the amount of departure from uniformity of motion of B 
relatively to A. In Lick Observatory Bulletin 208 I published 
a list of systems specially suited to the application of this method 
and urged the desirability of measuring them systematically. 

The late Lewis Boss deduced the mass ratios for a number 
of systems and published his results as an appendix to his 
Preliminary General Catalogue of Stars for 1900.0. Taking his 
values and a few obtained by other investigators and applying 
them to the masses for the systems, as given in Table I,f we have 
the following data on the masses of the components in visual 
binary systems : 

* Russell, Ap. Jour. 32, 363, 1910. 
f In the Appendix. 



THE KNOWN ORBITS OF THE BINARY STARS 
Table 12 



227 



Star 



n Cass 

40Erid. BC 

Sinus 

Procyon 

f Cancri AB 

6 Hydrae AB 

f Urs. Maj. Aa, Bb 

7 Virginis 

a Centauri 

£ Bootis 

THerc 

Melb4, AB 

n Here BC 

70 0ph 

Krueger 60 

86Pegasi 

* [K Pegasi AB (A is a Sp. Bin.) . 



Spec. 



F8 

A 

A0 

F5 

GO 

F8 

GO 

F0 

GO 

G5 

GO 

K2 

Mb 

KO 

Ma 

GO 

F5 



wii/wi 



0.76 
0.45 
0.39 
0.36 
1.00 
0.86 
1.00 
1.00 
0.85 
0.87 
0.45 
0.75 
0.80 
0.78 
0.67 
1.86 
0.60 



0.72 
0.44 
2.40 
1.17 
1.24 
2.02 
0.83 
1.19 
0.95 
0.73 
1.22 
0.67 
0.46 
0.80 
0.24 
0.35 
3.30 
(A+a) 



wii 



0.55 
0.21 
0.96 
0.43 
1.23 
1.74 
0.83 
1.19 
0.81 
0.63 
0.52 
0.51 
0.37 
0.72 
0.16 
0.65 
1.90 
-^J.T.K.] 



The evidence available at present leads to the conclusion that 
the fainter component in the system 85 Pegasi is the more massive 
one, but Boss considered the uncertainties so great that he 
finally gave the two components equal masses. The results for 
the other systems indicate that the fainter star is the less massive 
one, a conclusion which is in harmony with that derived from the 
spectroscopic binary stars. 

DYNAMICAL + PARALLAXES OF THE VISUAL BINARY STARS 

Equation (1), page 217, written in the form 
a 

where p is the parallax, may obviously be used' to compute the 
parallaxes of systems with known orbits for which a value of the 
masst can be determined or assumed from independent data; 
but the number of such systems is small and is not increasing 
very rapidly. It has, however, been known for many years 
that two or more sets of elements for a given system may have 
but little resemblance to each other as a whole and yet give 
* [IAck Obs. Bull. 483, Tab. I, #342 R, p. 22, Luyten, 1934.—J.T.K.] 
t The term "hypothetical" was used in earlier years to describe these 

parallaxes. 

J The Sun's mass is, as usual, taken as the unit in this discussion. 



228 THE BINARY STARS 

approximately the same value for the ratio a*/P 2 , which enters 
our equation. The reason is, as Jackson says, that the various 
sets of elements "give nearly the same arc for that portion of the 
orbit which has been observed and that this arc is sufficient to 
determine the gravitational attraction between the two stars." 
In fact, in order to define the relation between the parallax 
and the mass of a system, all that is needed is an observed arc 
long enough for the computation of a satisfactory value of the 
double areal velocity p 2 (ddldt). 

Comstock* was the first, apparently, to apply this principle, 
using it to derive the masses of systems in slow orbital motion 
for which parallaxes were available. Russell, f Hertzsprung,J 
and Jackson and Furner§ later developed formulas for deter- 
mining dynamical parallaxes on the basis of assumed average 
values for the mass m x + m 2 . Hertzsprung, in 1911, adopting 
the value (mi + w 2 ) = 1, derived "minimum hypothetical" 
parallaxes and concluded that, statistically, the ratio of the 
true to the minimum hypothetical parallaxes, does not vary 
greatly and can be expressed by 

log-^- = +0.27 ± 0.14 

Ph.min. 

Jackson and Furner, the first to publish an extensive list of 
dynamical parallaxes, used (ra x + m 2 ) = 2 for systems with 
orbits, and l/\/mi -f- m 2 = 0.855 for the much larger number 
with observed arcs too short for the computation of even pre- 
liminary elements. 

In 1923, Russell, Adams, and Joy^[ published their comparison 
of 327 dynamical parallaxes, computed by Russell on the assump- 
tion that the mass of each system equals that of the Sun, with 
the spectroscopic parallaxes derived at Mount Wilson. They 
grouped the stars according to spectral class, separating giants 

*Publ. Washburn Obs. 12, 31, 1908. 

t Determinations of Stellar Parallax, A.J. 26, 147, 1910; Science (N.S.) 
34, 523, 1911. 

t Vber Doppelsterne mil eben merklicher Bahnbewegung, A.N. 190, 113, 
1911. 

§ Jackson and Furner: The Hypothetical Parallaxes of 556 Visual Double 
Stars, with a Determination of the Velocity and Direction of the Solar 
Motion, Mon. Not. R.A.S. 81, 2, 1920. 

If A Comparison of Spectroscopic and Dynamical Parallaxes, Publ. A.S.P. 
36, 189, 1923. 



THE KNOWN ORBITS OF THE BINARY STARS 229 

from dwarfs from F6 to M, taking absolute magnitude = 3.0 
as the dividing point. They also computed the absolute magni- 
tude for each pair, both from the dynamical and from the 
spectroscopic parallaxes. The relationship between absolute 
magnitude and mass was striking; stars of all spectral classes 
and giants as well as dwarfs, "fell into line." "It is evident," 
they wrote, "that statistically considered, the mass of a binary 
system is a function of its absolute magnitude." The theoretical 
explanation of the mass-luminosity relationship, has been given 
by Eddington. In his more recent work on dynamical parallaxes, 
Russell* has developed formulas which include a factor, n, 
depending upon the mass-luminosity relationship and has drawn 
up tables from which its value may be obtained. 
For pairs with orbits, his formula is 

p = ndx = naP~ H 
and for physical pairs with only a short observed arc, he writes 

h = nh lf hi = l^£ = 0.418^^ 

where 8 is the observed distance, w the observed relative motion 
in seconds of arc a year, and I a factor (derived from a statistical 
discussion not given in detail) which is designed to make the 
mean values of d x and h x equal. The values of d x and hi (except 
for the factor of proportionality) are derived directly from the 
data of observation. To find n, three tables, A, B, and C, are 
provided. Table A gives the correction to be applied to the 
observed visual magnitude of a pair to reduce it to the "stand- 
ardized bolometric magnitude to." This correction, of course, 
depends upon, and varies with, the spectral class (that of the 
primary star is taken in general) or assumed temperature. The 
bolometric absolute magnitude, M 1} then follows from 

M i = to + 5 + 5 log d x (for pairs with orbits) 
« to + 5 + 5 log hi (for other physical pairs) 

With M x as argument, take out n from Table B, and with Mi 
and AMi the difference of magnitude between the two com- 
ponents, as arguments, take A from Table C. Then 

n = rioA. (1) 

* On the Determination of Dynamical Parallaxes, A. J. 38, 89, 1928. 



230 



THE BINARY STARS 











Table A 








Sp. 


Temp. 


Corr. 


Sp. 


Temp. 


Corr. 


Sp. 


Temp. 


Corr. 


BO 


23000° 


-0?9 


gGO 


5600° 


0?0 


dGO 


6000° 


+0?1 


B5 


16000 


-0 .1 


gG5 


4700 


-0.4 


dG5 


5600 


.0 


AO 


11000 


+0.2 


gKO 


4200 


-0.7 


dKO 


5100 


-0 .2 


A5 


8600 


+0 .3 


gK5 


3400 


-1 .5 


dK5 


4400 


-0.6 


FO 


7400 


+0 .3 


gM2 


3100 


-2 .0 


dMO 


3400 


-1 .5 


F5 


6500 


+0 .2 


gM7 


2700 


-2 .6 









Table B 



Mi 


n 


Mi 


Wo 


Mi 


Too 


-6 


0.065 





0.417 


6 


0.880 


-5 


0.097 


1 


0.496 


7 


0.961 


-4 


0.132 


2 


0.575 


8 


1.046 


-3 


0.189 


3 


0.652 


9 


1.138 


-2 


0.251 


4 


0.728 


10 


1.238 


-1 


0.337 


5 


0.S03 


11 


1.345 



Table C 



AM 


Mi = -4 


-2 





+2 and fainter 





1.00 


1.00 


1.00 


1.00 


1 


1.02 


1.01 


1.01 


1.01 


2 


1.05 


1.04 


1.03 


1.03 


3 


1.08 


1.07 


1.06 


1.05 


4 


1.11 


1.09 


1.08 


1.07 


5 


1.13 


1.12 


1.11 


1.10 


6 


1.14 


1.13 


1.12 


1.12 


7 


1.15 


1.14 


1.14 


1.14 


8 


1.16 


1.15 


1.15 


1.15 


9 


1.16 


1.15 


1.15 


1.16 


10 


1.16 


1.16 


1.16 


1.18 


12 


1.17 


1.17 


1.18 


1.19 



In 1929, Russell and Miss Charlotte E. Moore published the 
dynamical parallaxes of 1,777 double stars,* derived by the 
method just described, and in 1933, Miss Moore computed the 
dynamical parallaxes of 323 of the pairs of my own discovery, 
from observational data which I provided, f 

* A.J., 39, 165, 1929. 

t Lick Obs. BvU. 16, 96, 1933, #451. 



THE KNOWN ORBITS OF THE BINARY STARS 231 

Mr. R. 0. Redman,* in a paper published just before Russell's, 
also gave the results of an investigation planned to improve 
dynamical parallaxes by making the masses conform to the 
mass-luminosity relation instead of adopting a mean standard 
mass. His discussion differs from Russell's in its details and 
results in more laborious processes for the actual computations 
of parallax. Finsenf has recently made an instructive compari- 
son of the formulas developed by Comstock, Jackson, Hertz- 
sprung, and Russell, and has added a formula for use in the case 
of binaries whose periods are known, though only a small actually 
observed arc is available. Several pairs of this kind are given 
in double star catalogues. 

Dynamical parallaxes are, statistically, of a high order of 
accuracy, particularly when the mass-luminosity relation has 
been taken into account, but it does not follow that the dynamical 
parallax of an individual star can be taken as the measure of 
its distance. Comparison of the dynamical parallaxes computed 
by Russell and Miss Moore with the parallax values entered in 
Table I,J for example, shows that the mean of the differences is 
only +0"00016, but individual differences range from +0"009 
to — 0"008, with a few, for the nearer stars, that are much 
larger. 

DENSITY OF THE BINARY STARS 

Although every short-period spectroscopic binary star would 
be an eclipsing binary to an observer in the plane of its orbit, 
as Stebbins and others have remarked, the methods outlined in 
Chap. VII to determine the density of an eclipsing binary 
from the orbit elements cannot be applied to spectroscopic 
binaries in general, and still less to the visual binaries. But 
from the relations that have been established theoretically 
between the absolute magnitude, diameter, and temperature of 
a star, confirmed as they have been in many critical cases by 
interferometer measures and by the investigations on eclipsing 
binaries, it is possible to determine the diameters and the masses 
of stars, and hence their densities. 

* A Statistical Study of the Effect of the Mass-Luminosity Relations on 
the Hypothetical Parallaxes of Binary Stars, Mon. Not. R.A.S. 88, 33, 1927. 

fThe Determination of Dynamical Parallaxes of Double Stars, Mon. 
Not. R.A.S. 92, 47, 1931. 

% Appendix. 



232 THE BINARY STARS 

The fact that a star is a component of a binary system has no 
bearing either upon the principles involved in these methods, 
or upon their application and for this reason they are not treated 
here. It may be remarked, however, that the evidence now 
favors the conclusion that the densities of stars of the main 
sequence, from Class B to Class G, whether they are single 
stars or components in a spectroscopic or visual binary system, 
are very similar to those found for the eclipsing binaries. There 
are, of course, a few exceptions among components of binary 
systems, as, for example, the very dense companion to Sirius 
which will be discussed in detail in Chap. IX. 

MULTIPLE STARS 

In 1781, Herschel noted that the brighter star of the 5" pair, 
f Cancri, discovered by Tobias Mayer in 1756, was itself a 
double star with an angular distance of only 1" between its 
nearly equal components. In the years that have followed, a 
large number of such triple systems, and not a few that are 
quadruple, or multiple, have become known. During the Lick 
Observatory double star survey, for example, I catalogued at 
least 150 such systems previously unknown, and Prof. Hussey's 
work yielded a proportionate number. The triple was formed, 
in more than half of these cases, by the discovery of a close 
companion to one of the components of a wider pair previously 
catalogued by other observers, and in some cases there is no 
question but that the closer pair had been overlooked at the 
earlier date because it was below the resolving power of the 
telescope. 

The spectrograph has also revealed many triple and mul- 
tiple systems; sometimes, as in 13 Ceti or k Pegasi, by show- 
ing that one component of a visual binary is itself a binary 
too close to be seen as such with the telescope; again, as in 
Algol, by showing that the short-period spectroscopic binary 
revolves in a larger orbit with a third invisible star. 

In 1918 I estimated that 4 or 5 per cent of the visual binaries 
are triple or quadruple systems if components known only 
spectroscopically or through the perturbations they produce are 
included as well as visible additional companions. The compila- 
tion of the New General Catalogue of Double Stars offered oppor- 
tunities to check that estimate. I did not make a complete 
count, but was confirmed in my opinion that this estimate is 



THE KNOWN ORBITS OF THE BINARY STARS 233 

very little, if at all, too liberal. It seems to be a general rule 
that the distance between the components of the close pair 
in such systems whether visual or spectroscopic is small in 
comparison with that which separates the pair from the third 
star. However, there are exceptions to the rule. Thus we 
have in Hu 66, BC = 0'.'34, A and BC (= 02 351) = 0'/65; in 
A 1079, AB = 0-'23, AB and C = 0'/48; in A 2286, AB = 0"34, 
AB and C = 0'-'94; in A 1813, AB = 0'.'20 AB and C = 0'.'70; 
and in Hu 91, BC = 0"15, AB (= OS 476) = 0"54. Some 
allowance must, of course, be made for the effect of perspec- 
tive; the orbit plane of the closer pair may not coincide with 
the plane in which the third star revolves. But it is unlikely that 
this will modify the relative apparent angular distances greatly. 

The system of Castor affords an extreme example of the 
contrasting distances between the close and wide pairs in a 
quadruple star; each component of the visual pair is a spectro- 
scopic binary, the revolution periods being respectively three 
and nine days while the period of the orbit described by these 
two pairs is certainly greater than 300 years! The motion of 
the third star with respect to the closer pair in a triple visual 
system has in no instance been observed over an arc long enough 
to permit the computation of a reliable orbit. 

The various conclusions drawn in the present chapter as to 
masses and densities of the binary stars and as to the relations 
between the orbit elements, rest upon comparatively small 
numbers of pairs, but some of them, none the less, may be 
accepted as definitely established. Others may require modifica- 
tion, when additional data become available. 

References 

From the many papers published in recent years on subjects discussed 
in the present chapter, I have selected the following, in addition to those 
quoted in the footnotes, as representative. 
Ludendorff, H.: Weitere Untersuchungen liber die Massen der spektro- 

skopischen Doppelsterne, A.N. 211, 105, 1920. 
Hektzsprung, E. : On the Relation between Mass and Absolute Brightness 

of Components of Double Stars, B.A.N. 2, 15, 1923. 
Jackson, J. : The Masses of Visual Binary Stars of Different Spectral Types, 

Mon. Not. R.A.S. 83, 444, 1923. 
Shajn, G. : The Movement of the Line of Nodes in Spectroscopic Binaries 

and Variables and its Consequence, Ap. Jour. 67, 129, 1923. 
Eddington, A. S.: On the Relation between the Masses and Luminosities 

of the Stars, Mon. Not. R.A.S. 84, 308, 1924. 



234 THE BINARY STARS 

Brill, A.: Strahlungsenergetische Parallaxen von 123 visuellen Doppel- 

sternen, Veroff. Univ. Sternwarte Berlin-Babelsberg, 7, I, 1927. 
Lundmakk, K. : Statistical Concerning the Binary Stars, Ark. f. Mat. Astr. 

och. Fysik, 20A, 12, 1927. 
Pitman, J. H.: The Masses and Absolute Magnitudes of Binary Stars, A. J. 

39, 57, 1929. 
Shajn, G.: On the Mass-Ratio in Binary Stars and the Hypothesis of a 

Secular Decrease of Mass, A.N. 237, 57, 1929. 
Krbiken, E. A.: Many papers: e.g., Man. Not. R.A.S. 89, 589, 1929; B.A.N. 

4, 239, 1928; 6, 71, 1929; A.N. 238, 373, 1930. 



CHAPTER IX 

SOME BINARY SYSTEMS OF SPECIAL INTEREST 

Having studied the orbit elements of the binary stars in 
their more general relations, it will be of interest next to con- 
sider the various systems in themselves, the extent, and the 
limitations, of our knowledge of their motions and physical 
conditions. Selection is here an obvious necessity, and in 
making my choice I have been influenced in part by the his- 
torical associations connected with certain systems, in part 
by the peculiarities of the orbit. Some of the systems are 
among those for which our knowledge is relatively full and 
exact; others present anomalies still more or less baffling to 
the investigator. 

a CENTAXJRI 

Our nearest known stellar neighbor, a Centauri, is a system 
of more than ordinary interest. One of the first half dozen 
double stars to be discovered — the very first among the stars 
of the southern heavens — it also divides with 61 Cygni the 
honor of being the first whose approximate distance, or par- 
allax, became known. It consists of two very bright stars, 
0.3 and 1.7 magnitude, respectively, which revolve in a strongly 
elliptic orbit so highly inclined to the plane of projection that at 
times they are separated by fully 22", at others by less than 2". 

Accurate micrometer measures of relative position begin 
only with Sir John Herschel, in 1834, but meridian circle obser- 
vations date back to Lacaille's time, 1752. Since these early 
dates the system has been observed regularly with meridian 
circle, micrometer, and heliometer, and the position of its com- 
ponents has been measured on photographic plates. The 
material is therefore ample for a very good determination of 
the orbit elements and of the proper motion of each component 
and excellent use has been made of it by Roberts, See, Doberck, 
Lohse, and, in 1926, by Finsen, whose elements are quoted in 
Table I. Finsen's period is 1.25 years longer than that found by 

235 



236 THE BINARY STARS 

Lohse, but his other elements differ little from the earlier set and 
we may regard his results as definitive. The parallax is known 
with equal precision; the value resulting from the excellent 
heliometer measures by Gill and Elkin having been confirmed 
by later discussions of meridian observations by Roberts and 
others and by Wright's results from measures of the relative 
radial velocities of the components, to which reference has been 
made on an earlier page. Schlesinger* gives the value 0"760 
and assigns the proper motions — 3"604 in right ascension and 
H-0"739 in declination to the system of the two stars. The 
spectrograph has also given us the radial velocity! of the center 
of mass of the system, —22.2 km/sec. 

Taking Finsen's elements and the parallax value just quoted, 
we find that the semimajor axis of the system is 23.2 A.U., 
but since the eccentricity of the orbit is 0.52, the distance between 
the components at periastron is but 11.2 A.U. only a little 
greater than Saturn's mean distance from the Sun, whereas at 
apastron it is 35.3 A.U., a value about midway between those of 
Neptune and Pluto from the Sun. 

In 1904, when Wright measured the radial velocities of the two 
components of a Centauri, he found that the brighter star was 
approaching the Earth with a velocity of 19.10 km/sec, and the 
fainter one, with a velocity of 24.27 km/sec. That is, relatively 
to the center of mass of the system, the primary was receding from 
the Earth, the companion approaching it. The companion's 
position angle at that date was approximately 207°, and the 
nodal point of the orbit is 25?4; hence, on the system of notation 
we have adopted (Chap. IV), the sign of the inclination is 
negative.! 

The mass of the system corresponding to the adopted values 
of the parallax and orbit elements is 1.96 times that of the 
Sun, and all investigators of the proper motions of the two 
components agree that the brighter star is very slightly the more 
massive of the two. Besides being practically equal to the 
Sun in mass, it belongs to the same spectral class (GO) and has 
nearly the same absolute magnitude, 4.73 as compared with 
4.85. It is therefore almost a replica of the Sun. The fainter 
component belongs to Class K5, and its absolute magnitude is 

* Catalogue of Bright Stars, p. 110, 1930. 

t Moore, Publ. Lick Obs. 18, 115, 1932. 

% It was incorrectly stated to be positive, in the first edition. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 237 

6.10. Combining the values for proper motion and radial 
velocity, we find a space velocity of 25.2 km/sec, but little greater 
than that of the velocity of our Sun. 

A new chapter in the story of this system was opened when 
Innes, in 1915, discovered that "Proxima Centauri," a faint 
star of 10.5 apparent magnitude (15.0 absolute) had approxi- 
mately the same proper motion as the bright star, although a 
little more than 2° distant from it. It proved also to have very 
nearly the same parallax, and is no doubt physically connected 
with the two bright stars though too distant to affect their orbital 
motions. "Proxima" is a little nearer to the Sun (4.16 light 
years) than a Centauri (4.30 light years) and is therefore our 
nearest known stellar neighbor. 

snuus 

Several references have been made to Sirius on the earlier 
pages of this volume, but it will not be amiss to give a more 
connected account of the star here. It was in 1834 that Bessel 
noticed that the proper motion of Sirius, the brightest star in 
the sky, was variable. Six years later he noted a similar 
phenomenon in the proper motion of Procyon, and by 1844 he 
had worked out the nature of the variation sufficiently to 
become convinced that it was due in each instance to the attrac- 
tion of an invisible companion. His famous letter to Humboldt 
on the subject has often been quoted: "I adhere," he wrote, 
"to the conviction that Procyon and Sirius are genuine binary 
systems, each consisting of a visible and an invisible star. We 
have no reason to suppose that luminosity is a necessary property 
of cosmical bodies. The visibility of countless stars is no argu- 
ment against the invisibility of countless others." 

Peters examined the existing meridian circle observations in 
1851 and concluded that they supported Bessel's hypothesis; 
ten years later, T. H. Safford repreated the investigation and 
"assigned to the companion a position angle of 83-8 for the 
epoch 1862.1." The most complete discussion, however, was 
that of Auwers, who "placed the question beyond doubt by 
determining the orbits and relative masses of the bright star 
and the invisible companion; but before the results were pub- 
lished, Mr. Alvan G. Clark discovered the companion, in 1862, 
near its predicted place." Bond's measures for the epoch 



238 THE BINARY STARS 

1862.19, in fact, placed the companion 10'.'07 from the primary 
in position angle 84- 6. 

Since that time it has described more than an entire revo- 
lution and the orbit elements, now known with high precision, 
agree as well as could reasonably be expected with Auwers's, 
computed before the companion's discovery. Thus, Volet's 
orbit, computed in 1931, which differs very little from my own, 
published in 1918, has the revolution period 49.94, whereas 
Auwers gave 49.42 years. 

The eccentricity of the true orbit is greater than that for the 
orbit of a Centauri, but the inclination of the orbit plane is 
considerably less and the apparent ellipse is therefore a more 
open one, the maximum apparent separation of the components 
being about 11"2 and the minimum a little less than 2". The 
bright star is so exceedingly brilliant, however, that it is impossi- 
ble to see the faint companion with any telescope when it is near 
its minimum distance. Thus, periastron passage occurred 
early in 1894, but the last preceding measure was Burnham's 
in the spring of 1892 when the angular separation was 4'' 19, 
and the little star was not again seen until October, 1896, when 
my first measure gave an apparent distance of 3" 81. The 
photometric magnitude of Sirius (and also its photographic 
magnitude, since it is a star of Class AO) is —1.6. Estimates 
of the apparent magnitude of the companion are subject to great 
uncertainty because they are all affected by the presence of the 
intensely brilliant primary. Visual observers, in recent years, 
have adopted the value 8.5, and Wendell's value from measures 
with a double-image Rochon prism photometer* is 8.44. This 
value has recently been confirmed by Kuiperf from measures 
made with the Leiden 10-in. refractor, fitted with objective 
gratings. On the other hand, Vyssotsky,J using the 26-in. 
refractor of the McCormick Observatory and (a) coarse objective 
gratings, and (6) a rotating sector, derives the value 7.1 for the 
photovisual magnitude. 

The question of the precise magnitude of this star is an impor- 
tant one, because of its relation to the question of the star's 

* Harvard Annals 64, No. VI, 1909. 

t B.A.N. 6, 197, 1932. More recently, Kuiper using the 12-in. refractor 
at the Lick Observatory and suitable gratings, found the value 8.42, in full 
agreement with his Leiden result (Publ. A.S.P. 46, 99, 1934). 

t Publ A.S.P. 42, 155, 1930; Ap. Jour. 78, 1, 1933. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 239 

density. The mass of the system corresponding to the parallax 
and orbit elements in Table I is 3.36 times that of the Sun and 
this, with Boss's value for the mass ratio,* 0.39, gives 0.94O 
for the mass of the companion. The presence of the bright 
primary makes accurate determination of the spectral class 
difficult, but Adams finds that it is a little earlier than F0, and 
it is now usually called A7. It is the best known representative 
of the peculiar group called "white dwarfs." Taking the 
apparent magnitude as 8.4, and assuming the companion to 
have the surface temperature of a normal star of its spectral 
class, the density is found to be nearly 50,000 times that of water. 
The density corresponding to Vyssotsky's apparent magnitude 
determination, 7.1, is only about one-sixth as great. 

It was pointed out by several writers a number of years ago 
that a star of great density should produce an "Einstein shift" 
in its spectrum far greater than the one predicted in the spectrum 
of the Sun. In a single star this prediction cannot be tested 
observationally for there is no way to distinguish between an 
Einstein shift and the shift arising from the star's radial velocity. 
In a double star system, like that of Sirius, however, for which 
we know the parallax, the orbit elements and the mass ratio, 
the radial velocity of the faint component for any date can be 
computed from the observed radial velocity of the primary. 
Allowance can then be made for it and any residual Einstein 
shift detected. The brilliance of Sirius makes the observations 
of the spectrum of its companion extremely difficult, as I have 
already noted, but Adams f with the 100-in. reflector at Mount 
Wilson secured spectrograms, the measures of which, after 
allowance for the radial velocity and the blend effect from the 
scattered light of the bright star, gave a displacement correspond- 
ing to +19 km/sec, and Moore, J using the Mills spectrograph 
attached to the 36-in. Lick refractor, later found precisely the 
same displacement. The theoretical value, according to Edding- 
ton, is +20 km/sec. 

These observations evidently support the greater value for 
the density of the companion and hence the value 8.4 for the 
apparent magnitude, but further investigation is desirable. 

* Preliminary General Catalogue, p. 266, 1910. 
t Proc. Nat. Acad. Set. 11, 382, 1925. 
t Publ. A.S.P. 40, 229, 1928. 



240 THE BINARY STARS 

Accepting the value 8.4 for the apparent magnitude, it follows 
that the primary is 10,000 times as bright as its companion, 
whereas it has only two and a half times its mass and is less 
than 1/50,000 part as dense. And yet the two stars, presumably, 
had a common origin. No satisfactory theory has been advanced 
to account for such a system. If we adopt Vyssotsky's value, 
7.1, the contrasts are not so great, but the difficulties in the way 
of an explanation are not lessened. 

There is another question relating to this system that must be 
considered: Is a third body present? Fox, on one night in 1920, 
suspected one less than a second of arc from B, the Clark com- 
panion, and van den Bos, Finsen, and other observers at the 
Union Observatory felt certain that they saw such a companion, 
about a magnitude fainter than B, on several nights in 1926, 
1928, and 1929. On other nights, however, when they rated the 
definition as equally good, they did not see it, and Innes, in 
publishing the observations,* added that "in view of the negative 
evidence, it would be wrong to assert that the companion C 
exists without any doubt." 

On the negative side, it is also to be said that neither Burnham 
nor Barnard, both noted for remarkable keenness of vision, saw 
such a companion at any time in the course of their many meas- 
ures of the system with the 36-in. Lick and 40-in. Yerkes refrac- 
tors. My own eyes are not so keen as theirs were, but I must 
add that I have never seen one, though I have measured the 
system annually since 1896, often under the most favorable 
observing conditions. Although negative evidence is always 
less convincing than positive testimony, I think it fair to return 
the Scotch verdict "not proven" on the question of the direct 
observation of this companion. 

There is, however, another method of investigating the 
problem, and that is to ascertain whether the measures of the 
system AB give evidence of perturbations that might be explained 
by a third body. Zagarf and Volet, t to name only the two 
most recent writers on the subject, have made thorough investiga- 
tions using different methods and both agree not only that such 

* The Observatory 52, 22, 1928. 

f F. Zagar, II terzo corpo nel sistema Sirio, R. Oss. Astr. Padova Nr. 23, 
1932. 

% Ch. Volet, Recherche des perturbations dans le systeme de Sirius, 
Bull. Astron. 8, 51, 1932. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 241 

perturbations exist, though of small amplitude, but also give 
approximately the same hypothetical orbit for the disturbing 
body, on the assumption that it is a companion to B. Unfor- 
tunately, the computed position angles of C, the third body, 
differ from those observed in 1920, 1928, and 1929 by amounts 
ranging from 70° to 170°. The 1926 observation is the only 
one in reasonable agreement with the theory. It is, however, 
impossible to say definitely, as Volet points out, that the hypo- 
thetical close companion attends component B; it may attend 
the brilliant primary A, in which event it would be quite impossi- 
ble to observe. 

KRUEGER 60 

The system known as Krueger 60, though the closer pair was 
really discovered by Burnham in his careful examination of all the 
double stars noted by Krueger in the course of his meridian 
circle observations, offers a strong contrast to the two we have 
been considering, not only in its appearance but in many of 
its physical characteristics, but like them it is remarkable for 
its large proper motion and its large parallax. Sirius and 
a Centauri are two of the brightest stars in the sky and are 
also of great absolute brilliance; Krueger 60 is only of the ninth 
magnitude, despite its large parallax, and is among the feeblest 
of known stars in its actual radiating power. 

The companion has now made very nearly one complete revolu- 
tion since its discovery in 1890, and since 1900, when Doolittle 
called attention to the rapid orbital motion, has been very well 
observed, Barnard, in particular, making a remarkable series of 
measures — often on 18 to 21 nights a year — extending from 1900 
to 1921. The orbit elements, therefore, while not so accurate 
as those of Sirius, are very well known. The parallax of the 
system is even better known, and the proper motion is also 
accurately determined. 

It happens that several independent stars are in the same 
field of view with Krueger 60, one, of 9.6 magnitude, being less 
than 30" distant in 1890, and but little over 1' now. Since the 
apparent orbit of the binary is a fairly open one, the minimum 
angular distance between the components exceeding 1"4, the 
conditions are specially favorable for the determination of the 
mass* ratio of the two components (see Chap. VIII). Our 
knowledge of the physical conditions in the system is therefore 



242 THE BINARY STARS 

far more complete than is the case for the average system with 
good orbit elements. 

The parallax, 0"257, and my orbit elements computed in 1925 
(P = 44.27 years, a = 2 / '46)give the mass 0.45 O ; using Huffer's 
later elements (P = 44.52 years, a = 2''362) and the same 
parallax, the computed mass is 0.40 O. These are probably the 
limiting values. Alden,* in 1925, found the mass ratio to 
be 0.835. With Huffer's elements, this gives the mass of A 
as 0.220, that of B, 0.18O. The latter, which is probably 
correct to within one or two units of the second decimal place, 
is the smallest mass so far established for any star. 

Estimates of the magnitudes, as is usual in double star systems, 
vary considerably. The combined magnitude is given as 9.0 
in the BD, and 9.1 in the Astronomische Gesellschaft Catalogue, 
but on the International (photovisual) Scale, Kuiper finds it 
to be 9.64, and the difference of magnitude of the two components, 
1.56. This would give 9.9 and 11.4 as the apparent magnitudes 
of A and B, respectively. The corresponding absolute magni- 
tudes are 11.9 and 13.4. According to Adams, the spectrum of 
A is Mb, and it is probable that the spectrum of B is of even 
later type. Both stars therefore belong to the class of red 
dwarfs, and the fainter one is one of the least luminous stars 
known. We may agree with Russell that they are nearing "the 
very end of their evolutionary history." 

It is worth noting that Krueger 60 is one of the eight known 
visual binary systems within five parsecs or about 16 light-years 
distance. It is because it is so near us, relatively speaking, that 
we know anything about it. The mean parallax for stars 
of apparent magnitude 9.0 is, according to Seares, 0"0039. 
Remove Krueger 60 to the corresponding distance (more than 
800 light years) and it would be beyond the resolving power of 
any existing telescope, and the combined image of the two 
components would be of about the eighteenth magnitude. 

61 CYGNI 

There are a number of wide double stars whose components 
have large proper motions of nearly the same amount. If the 
directions of the motion are different, the measures soon enable us 
to decide whether the pair is a physical, or, like 2634 (ADS 3864), 
merely an optical one. But when the directions differ but little 
* Pop. Astron. 33, 164, 1925. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 243 

and the apparent relative motion is small or nearly rectilinear, 
it is not always easy to classify the pair on the basis of the 
micrometer measures alone. Consider, for example, S1321 
(ADS 7251). The two components,* both 8.1 magnitude, have 
an angular separation of a little less than 20" and the relative 
motion in 92 years has been but 23° and 1"4. When the posi- 
tions of B with respect to A are plotted, they lie along a straight 
line within the error of measure. But the large proper motions 
differ so little that we may adopt the value 1"683 in 247*6 for 
both components, and the parallax, +0"165, is the same for 
each. Unquestionably they constitute a physical system. 
This is also true of 34 Groombridge (ADS 246), though the 
two components, separated nearly 40", differ in brightness by 
more than two magnitudes and their relative motion in 50 years 
is but 3?5 and less than 1"0, for they have the same large parallax, 
+0"282, and proper motion, 2V89 in 82°5. 

Other examples might be given (e.g., E2398 = ADS 11632t), 
but the best known pair of this class is undoubtedly 61 Cygni, 
(22758 = ADS 14636) which is famous also as the first star for 
which an approximately correct value of parallax was determined. 
This bright pair (magnitudes 5.57, and 6.28) has been known 
since the time of Bradley who, as quoted by Sir John Herschel, 
gave the position for the date 1753.8 as 54°36' nf 19''628. Her- 
schel listed the measures made to the end of the year 1822 and 
argued from them and from the large proper motion assigned to 
the pair by Piazzi and Bessel that the two stars constitute a 
binary system J as otherwise in the course of "nearly 70 years, 
during which they have been observed, one of them would 
doubtless have left the other behind, without supposing a coin- 
cidence too extraordinary to have resulted from accident." 
Nor did he fail to point out that the data make "61 Cygni a 
fit object for the investigation of parallax." 

This early argument notwithstanding, the question of the 
character of the pair long remained an open one, for it developed 
that the proper motions of the two components are not quite 
identical, Auwers giving the values 5" 191 in 51-52 and 5" 121 
in 53°68 for A and B, respectively, and the numerous observa- 
tions since 1830 for many years gave little evidence of departure 

* [+53° 1320/21, HD 79210/11, in U Ma.-J.T.K.] 
t [+59° 1915, in Draco.-J.T.K.] 

% Observations of 380 Double and Triple Stars in the Years 1821, 1822 and 
1823, by J. F. W. Herschel and James South, p. 367, London, 1825. 



244 THE BINARY STARS 

from relative rectilinear motion. C. F. W. Peters,* it is true, 
computed a set of orbit elements in 1885, finding a = 29'-' 48, 
P = 782.6 years, and e = 0.17, but the observed arc was entirely 
too short for accurate orbit computation and as late as 1905 
Burnham clearly intimated that he thought the pair an optical 
one. In that same year, however, Osten Bergstrand, from 
a thorough examination of all available data, including a special 
series of photographs taken by himself in the years 1899 to 1903, 
demonstrated the physical relationship of the two stars by show- 
ing not only that they have the same parallax, but also that 
the path of B relatively to A is concave, f This was confirmed 
by Schlesinger and Alter, in 19104 

More recently provisional sets of elements have been computed 
by P. Baize § and by Alan Fletcher, f The former, using graphical 
methods, finds a nearly circular orbit (e = 0.013) with a period 
of 756 years and a semimajor axis equal to 32 / -'8. The latter, 
adopting a mass value of 1.126 from the mass-luminosity relation- 
ship (Baize's elements give 2.28) and using special analytical 
methods, finds an orbit of marked eccentricity (e = 0.404) 
with a period of 696.63 years and a semimajor axis equal to 
24"525. This shows how wide the range of possible solutions 
still is. It is probable that the components are nearly equal 
in mass, but whether they are each equal to the Sun or have 
only half its mass is still in doubt. After another half century 
it will be possible to reach more definite conclusions. 

The system is, in any event, one of vast dimensions, its orbit 
having a semimajor axis at least twice as great as that of Pluto, 
but it must be emphasized that orbits of this size are by no means 
exceptional. There are certainly far more visual binaries with 
orbital dimensions of this order than there are of systems like 
S Equulei, which by way of contrast, we shall next consider. 

8 EQUULEI 

Until it had, recently, to yield its place to 531, to which 
Dawson in his preliminary orbit computation assigns a period 

* Bestimmung der Bahn des Doppelsterns 61 Cygni, A.N. 113, 321, 1886. 
t Untersuchungen tiber das Doppelsternsystem 61 Cygni, Nova Acta R. 
Soc. Set. Upsaliensis, Ser. IV, Vol. 1, n. 3, 1905. 
X See Chapter IV, page 121. 

§ Le Systeme double de 61 Cygni, Bull. Soc. Astron. de France 41, 20, 1927. 
f The Binary System 61 Cygni, Mon. Not. R.A.S. 92, 121, 1931. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 245 

4.66 years*, 5 Equvlei was the visual binary of shortest known 
period. This, of course, excludes Capella, which has been 
measured with the interferometer, and the pair £ Ursae Majoris 
Aa, one of whose components is invisible, but is known not 
only from the variable radial velocity of A but also from the 
perturbations it produces in the relative orbit of the pair AB. 

For many years, the period of 8 Equuhi was supposed to be 
11.4 years, and the orbit nearly circular, for the two components 
are nearly equal in brightness, and the pair is below the resolving 
power of existing telescopes at times of minimum angular 
separation. Hussey, in 1900, however, showed conclusively 
that the period is only half as long, 5.7 years, and the orbit a 
fairly elongated ellipse (e = 0.39). Another system resembling 
6 Equvlei in that the two components had been mistaken for 
each other after passing the point of minimum separation, is 
£ Scorpii AB. The accepted orbit, prior to 1905, had a period 
of about 105 years and an eccentricity of about 0.13, but my 
measures with the 36-in. refractor in 1904 and 1905 proved this 
to be incorrect. I found that the true period is only 44.7 years 
and the orbit really a very eccentric one (e = 0.75). 

Returning to S Equulei, we note that the orbit elements are 
well determined, the period in particular, since the pair has 
made 14 revolutions since its discovery by Otto Struve in 1852. 
The semimajor axis is 0"27 and this, since the paraUaxf is 0"066, 
corresponds to about 4 A.U. The orbit is therefore decidedly 
smaller than that of Jupiter and resembles the larger and more 
eccentric minor planet orbits. The mass of the system is 2.11 
times that of the Sun and the two components probably are 
nearly equal in mass. Each is therefore approximately equal 
to the Sun in mass, but since the absolute magnitudes of the two 
are, respectively, 4.4 and 4.5 and the spectral class is F5, they 
are brighter than the Sun and shine with a whiter light. 

TWO MULTIPLE SYSTEMS 

Two of the best known double stars in the northern heavens 
are Castor (a Geminorum), discovered by Bradley and Pound 

* See footnote, p. 205, for Kuiper's discovery of a visual binary with period 
of less than 2 years. 

t Schlesinger's Bright Stars. Maxwell (Publ. Lick Obs. 16, 311) found 
0- 048 from the orbit and the relative radial velocity of the components. 
This gives a = 5.6 A.U., which is a little larger than that of Jupiter. 



246 THE BINARY STARS 

in 1719, and £ Ursae Majoris, discovered by Sir William Herschel 
on May 2, 1780. Both pairs are among those used by Herschel 
in the papers, presented to the Royal Society in 1803 and 1804, 
in which he gave observational as well as theoretical reasons 
for his belief that many double stars are physical systems, with 
components in orbital motion. £ Ursae Majoris was also the 
first, and Castor perhaps the second pair for which an orbit was 
computed. Savary, in 1830, used the former system to illustrate 
his method of orbit computation,* and Sir John Herschel, in 
1833, used both pairs, among others, to illustrate his very 
different method. The orbital motion in £ Ursae Majoris is 
far more rapid than that in Castor, and for that reason these 
early orbits computed for it were more successful than Herschel's 
orbit of the latter pair. 

Struve began to measure £ Ursae Majoris in 1826 and from that 
time on the visual observations have been numerous and well 
distributed. The two components differ but one magnitude in 
brightness, and the angular distance between them, even at 
minimum, is but little less than one second of arc. The pair is 
therefore also well suited for photographic measurement, and, 
in fact, the photographic measures are more accurate than the 
visual ones, as appears from the series made at Potsdam by 
Hertzsprung and Munch in the years 1914 to 1923, and from the 
one by Przybyllok, at Konigsberg in the following years. 

Ultimately, an orbit of a system of this character based entirely 
upon photographic measures will probably be more accurate 
than one resting upon visual observations, but it is not so certain 
that modern photographic measures used in combination with 
earlier visual ones, particularly if the latter were made with small 
telescopes, will give a better orbit than visual measures alone, 
unless very great care is exercised in making the combination. 
Van den Bos did not overlook this point when, in 1928, he 
computed the orbit of £ Ursae Majoris, which has now described 
nearly two revolutions since Struve's first measures, and his 
elements, which differ but little from those of Norlund (1905), 
may be regarded as practically definitive. He gives P = 59.863 
years, e = 0.4128, and a = 2"5355. 

If this were a simple binary system there would be no occasion, 
other than its association with the early methods of orbit com- 
putation, to single it out for special notice. It is, however, not 

* See Chap. IV. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 247 

a binary, but a quadruple system. Wright* in 1900, noted that 
the radial velocity of the brighter star is variable, and Norlund, 
in his investigation of the orbit of the visual pair in 1905, dis- 
covered a perturbation with an amplitude of 0"05 and a period 
of about 1.8 years. This he attributed to the presence of a 
third body, but he was apparently unaware of Wright's discovery. 
Wright, in 1908, showed that the radial-velocity observations 
confirmed Norlund's period, for which van den Bos gives the more 
precise value, 1.8321 years. In a sense, as Hertzsprung has 
remarked, this may be called the shortest period established for 
a binary star on the basis of micrometrical observations, f This 
short-period orbit proves to be elliptic, (e = 0.50) and to be 
highly inclined to the plane of projection, the inclination, 
whose sign is fixed by the radial velocity measures, being — 84°5. 
The unseen companion, a, is therefore in retrograde motion, 
about A, just as B is. Apparently, the general rule is that the 
wide and close pairs in a triple or quadruple system revolve in 
the same direction, but this is not without exception. It is 
not followed in four of 21 systems which van den Bos examined. 

In 1918, the Lick observers announced that the fainter com- 
ponent, B, of the visual system is also a spectroscopic binary of 
short period. BermanJ has recently made a thorough investiga- 
tion of the orbit of this pair and finds it to be circular, with a 
revolution period of only 3.9805 days. The orbital inclination 
is probably too small to permit even a partial eclipse, and van 
den Bos finds that, notwithstanding its high inclination, the 
pair Aa is also not an eclipsing binary. 

Van den Bos investigated the question of the distribution of 
masses in this system; with the advantage of the knowledge of 
the orbit of Bb, Berman has repeated the investigation. Taking 
the value 0"126, the mean of the 10 modern results, for the 
parallax, and van den Bos's elements of the visual pair, he 
finds the total mass to be 2.27 times that of the Sun. Adopting 
0.77 as the best value the present data will give for the mass 
ratios, the mass of Aa is 1.280, of Bb, 0.990. Taking van den 
Bos's value for the mass ratio a/A he finds the mass of A to be 
0.93 O , that of a, 0.35 O . The mass ratio b/B may have, accord- 
ing to Berman, any value between x /z and }-£. 

* Ap. Jour. 12, 254, 1900. 

t Prior to 1935. 

i Lick Obs. BuU. 16, 109, 1931, #432. 



248 THE BINARY STARS 

The combination of two such unlike short-period binaries 
as the components of a 60-year-period system presents a problem 
of unusual interest to the student of the origin of the binary 
stars, as Berman has not failed to remark. 

Turning next to Castor, we find an even more remarkable 
multiple system, for not only are the bright stars, A and B, of 
the historic double, the visible members of short-period spec- 
troscopic binaries, but a third star, of the ninth magnitude, 
73" distant in 165°, which has the same parallax and proper 
motion as the bright pair, is also a short-period binary. The 
system Bb was discovered by Belopolsky, at Pulkova, in 1896, 
the system Aa, by Curtis, at Lick, in 1904. Curtis's investiga- 
tions* showed that B and its faint companion revolve in prac- 
tically circular orbits in a period of 2.928285 days, whereas A 
and its invisible companion travel in elliptic orbits (e = 0.5033) 
and have a period of 9.2218826 days. Curtis, moreover, on the 
basis of the observed data and what seemed to be reasonable 
assumptions, concluded that the fainter system Bb was about 
six times the more massive one. Luytenf has recently given 
reasons for adopting the more moderate mass ratio of two to 
one as the maximum possible, but even that is an anomalous 
result, for there are few exceptions (and those not above suspi- 
cion) to the rule that in binary systems, both visual and spectro- 
scopic, the brighter component is the more massive one. Luyten, 
himself, regards the true mass ratio as still unknown, and a 
careful review of all of the published data leads me to the same 
conclusion. 

In 1920, Adams and Joyt announced the fact that the ninth- 
magnitude companion to Castor, which we may call Castor C, 
is a short-period binary, and in 1926 Joy and Sanford§ computed 
its orbit from the measures of radial velocity. Before their 
paper was sent to press, van Gent's paperlf appeared, with an 
orbit of the pair as an eclipsing variable based upon photometric 
observations. The combined data show that the components 
of Cc revolve in circular orbits with an inclination of 86°4 in a 
period of 0.814266 day. It also appears || that the brighter 

* Lick Obs. Bull. 4, 55, 1906. 

t Publ. A.S.P. 45, 86, 1933; Publ. Minn. Obs. 2 (1), 3-13, 1934. 

t Publ. A.S.P. 32, 158, 1920. 

§ Ap. Jour. 64, 250, 1926. 

1[ B.A.N. 3, 121, 1926. 

|| See Table 1, Chap. VII, p. 200. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 249 

component, C, has a mass 0.63 O, the fainter one, c, a mass 
0.57 O, that the densities of the two are, respectively, 1.40 and 
1.80 O, their radii, 0.76 O and 0.68 O and the distance between 
their centers 2.7 X 10 6 km. In other words, our knowledge of 
this latest known of the four orbital systems in this great multiple 
group, three spectrographic and one visual, is by far the most 
accurate and complete. 

Our knowledge of the first known of the group, the visual 
binary pair, on the other hand, is the most incomplete. The 
majestic scale of the relative orbit, with a semimajor axis nearly 
twice as great as the mean distance of Pluto from the Sun, is, 
of course, responsible. Many hundreds of measures have been 
made since 1826, when Struve began his series, but in the interval 
to 1931 the companion had described an arc of but 53°. The 
earlier measures, back to 1719, give only the direction of B 
from A, with occasional rough estimates of distance. It is 
now evident that the point of maximum separation in the third 
quadrant was passed some 25 or 30 years ago, and the range of 
possible orbit solutions is therefore much smaller than it was in 
1900, but orbits with periods ranging from 340 to 477 years 
still represent the latest measures about equally well. As so 
frequently happens, however, the mass of the quadruple system 
Aa-Bb is better known than the orbit elements. It is less than 
was thought some 20 years ago, and is probably about five 
times the mass of the Sun. 

It may be noted that while there is no question of the physical 
relationship of the distant system Cc to the bright quadruple 
system, there is no hope at all of securing evidence of its orbital 
motion about the center of gravity of Aa-Bb for many centuries, 
for, on the most favorable assumptions, the revolution period 
must be of the order of 25,000 years. 

CAPELLA 

In 1899, Campbell, at the Lick Observatory, and Newall, 
at the University Observatory, Cambridge, England, inde- 
pendently discovered the fact that Capella is a spectroscopic 
binary star, with the spectra of both components visible. Camp- 
bell's announcement was published in the Astrophy steal Journal 
for October, 1899, and Newall's, in the November, 1899, number 
of the Monthly Notices, R.A.S. In March 1900, Newall* pub- 

* Mon. Not. R.A.S. 60, 418, 1900. 



250 THE BINARY STARS 

lished a paper on the binary in which he showed that the period 
is approximately 104 days, that the two components are nearly 
equal in mass and not very unequal in brightness and argued 
that the inclination of the orbit exceeded 27°. Reese,* in 1901, 
published a more detailed investigation, based upon spectrograms 
secured at the Lick Observatory in the period September 1, 1896 
to September 27, 1900. His value for the period is 104.022 days, 
for the eccentricity, 0.0164, and for K, the semiamplitude of 
the velocity curve of the primary star, 25.76 km. He gave 
1.26 as the approximate value for the ratio K^/K 2 = M l /M 2 . 

Interesting as these facts are, they would not, taken alone, 
call for special comment. But Anderson, f in 1920, successfully 
applied the interference method of measurement developed by 
Michelson to observations of Capella, using a new type of 
interferometer which he had himself devised, and these measures 
were continued later in that same year and early in 1921 by 
Merrill who utilized the entire series for the computation of an 
orbit of the system, % adopting, however, Reese's period, which 
Sanford§ had shown to be very nearly correct. He found the 
orbit a little more nearly circular than Reese gave it, his value 
for e being 0.0086. 

Important results arose from the combination of the inter- 
ferometer with the spectrographic orbit data. This permitted 
the derivation of the values of a and *', separately, fixed the 
algebraic sign for i, and, most important of all, gave an extremely 
accurate value of the parallax of the system, and hence also 
values of the masses of the two components and of the separation, 
in linear measure, of their centers (ai + a 2 ). These values are: 

a = 0"05360, i = -41°08, (a x + a 2 ) = 126,630,000 km 
7T = 0"0632, mi = 4.2 O, m 2 = 3.3 O 

The visual magnitude of Capella is 0.21 and since the two 
components differ in brightness by 0™5, their magnitudes 1[ are, 
respectively, 0.74 and 1.24, which, with the given parallax, 
correspond to the absolute magnitudes —0.26 and +0.24. 
Both components are therefore giant stars. 

* Lick Obs. Bull. 1, 32, 1901. 
t Ap. Jour. 61, 257, 1920. 
t Ap. Jour. 56, 40, 1922. 
§ Publ. A.S.P. 34, 178, 1922. 

If According to the accurate formula, not the approximate values tabu- 
lated on p. 54. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 251 

The maximum angular separation of the components measured 
with the interferometer is 0"0550, a fact that fully accounts for 
the failures to separate the two components with the most power- 
ful telescopes. Interferometer measures, however, are made 
visually, and, in a sense, we may therefore refer to Capella as 
the visual binary of shortest known period. 

Merrill's value of the parallax of Capella is probably as accurate 
as any stellar parallax value we have. This fact, and the 
completeness and accuracy of our knowledge of the other data 
relating to the system, led Eddington to adopt the brighter 
component, which belongs to spectral class GO, as the foundation 
for the numerical application of his theory of the Internal Con- 
stitution oj the Stars, including the mass-luminosity relation. 
Assuming the effective temperature to be 5200°, Eddington finds 
the absolute bolometric magnitude of this component to be 
-CT36, which is 5.26 magnitudes brighter than the Sun. From 
these values he derives the total radiation of the star (127 X O), 
its radius (13.74 X O), and its mean density 0.00227 gm per 
cubic centimeter and then proceeds to further conclusions as to 
the star's internal structure and temperature, which do not 
come within the scope of this volume. 

THE MASSIVE BINARY HD 698 ( = BD. + 67°- 28) 

As long ago as 1904, Hartmann,* in examining the spectro- 
grams of 5 Ononis (Class B0), secured at Potsdam in the years 
1900 to 1903, discovered that the calcium line K differed from 
the other lines in the spectrum (mostly due to hydrogen and 
helium) in two important particulars. It was narrow and 
sharp, whereas the other lines are more or less diffuse, and it 
gave a nearly constant velocity, with a mean value about equal 
to the velocity of the center of mass of the system, instead of 
showing as the other lines did, variable velocity with a range of 
about 200 km/sec. The calcium line, apparently, did not 
originate in the star. 

In explanation, Hartmann assumed the existence of a calcium 
cloud stationary in space (at least so far as radial velocity is 
concerned) lying between us and the star. This was plausible, 
since the Orion region is -known to be one filled with nebulous 
matter and the mean velocity, +18.7 km/sec, from the calcium 
line was nearly the same as that of the Sun's velocity of recession 
from the Orion region. 
* Ap. Jour. 19, 268, 1904. 



252 THE BINARY STARS 

Questions, however, soon arose, for it appeared — particularly 
from Slipher's researches* — that the calcium lines in other stars 
of the Orion type, that is, of classes O to B2, widely distributed 
around the sky in low galactic latitudes, exhibited similar 
anomalous phenomena, and Miss Heger (Mrs. C. D. Shane), 
in 1919 and 1921, showed that, in some of these stars at least, 
the sodium D lines behaved in like manner. 

The theory now accepted to account for the observed facts 
is the one advanced by Eddingtonf in 1926. He postulated the 
existence of a uniformly distributed interstellar cloud of extremely 
low density, and showed that it was competent to produce the 
observed absorption in the spectra of all stars distant enough to 
give sufficient depth to the cloud. More recent evidence, 
based upon direct observation, has been given by O. Struve,J 
Gerasimovic,§ and particularly, Plaskett and Pearce,1f that 
practically establishes the fact of the existence of Eddington's 
hypothetical cloud. For it appears that the intensities of the 
interstellar lines increase with the distance of the stars, that the 
cloud, like the stars, rotates about the galactic center, and that 
the mean distance of the cloud is about half the average distance 
of the stars. 

One difficulty remained. While it was not to be expected that 
the interstellar lines of calcium and sodium would show in the 
spectra of stars of classes F to M, since they would be masked 
by the broad, diffuse stellar lines, it was hard to explain why 
they did not appear in the spectra of at least some of the binaries 
of Class A in which the range in variable velocity is large. 
Pearce's investigation of the system HD 698, of spectral class 
B9sek|| and visual magnitude 7.08 removes this difficulty. 

Merrill and Humason,** in the course of their study of stars 
with bright hydrogen lines, found the radial velocity of this 
star to be variable. Their announcement was made in 1925, 
and Pearce;}! has now computed the orbit of the pair, on the 
basis of spectrograms secured at the Dominion Astrophysical 

*LoweU Obs. Butt. 2, 1, 1909. 

t Bakerian Lecture: Diffuse Matter in Interstellar Space, Proc. Roy. Soc. 
A, 111, 424, 1926. 

t Ap. Jour. 66, 163, 1927; 67, 353, 1928. 

§ Ap. Jour. 69, 7, 1929. 

% Mon. Not. R.A.S. 90, 243, 1930. 

|| It may be well to explain that the letter s signifies sharp lines, the letter 
e, emission lines, and the letter k, the presence of interstellar calcium lines. 

** Ap. Jour. 61, 389, 1925. 

* Mon. Not. R.A.S. 92, 877, 1932. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 253 

Observatory. He finds the spectrum of the primary to be that 
of a star of high intrinsic luminosity for Class B9, and that of the 
secondary (apparently a normal star of Class B5) to be relatively 
faint and measurable only at times of maximum separation 
of the lines. The interstellar calcium absorption line, K, is also 
present and was measured on 18 of the 40 plates. 

Before commenting upon the K line let us note the orbit 
elements derived from the measures of the lines for the primary 
star. The period is 55.904 days, the eccentricity 0.033, neither 
value being in any way exceptional. The semiamplitudes of the 
velocity curves, too, while large (K = 85.5 km, K x = 215.5 km), 
are not extraordinary, a statement that may also be made of the 
mass ratio they give, m 2 /mi = 0.40. But the minimum mass 
values are remarkable, msm 3 i being 113.2 O, and m 2 sm 3 i, 
44.9 O. These are the largest mass values so far found in any 
binary system, and it is to be remembered that the inclination 
is less than 90°, since the pair is not an eclipsing variable, and 
that the true masses therefore are even greater than those 
given. 

Turning now to the calcium line K, we note that Pearce found 
a blend of the interstellar line with the stellar K line of the 
primary through about 41 days of the 56-day period, the blended 
velocities varying from +38 km/sec to -52 km/sec. But 
for a few days when the stars were passing through the nodes, 
the three lines, from the primary star, the secondary star, and 
the interstellar source, were clearly separated, and the interstellar 
lines from 12 observations at these favorable times gave a con- 
stant velocity of -13.9 km/sec. The velocity of the center 
of mass of the binary system is -24.5 km/sec. These results 
for a Class A star afford strong support for Eddington's theory 
of the origin of the interstellar lines in spectroscopic binary 
star spectra. 

ALGOL * 

Persei, or Algol, as it is more commonly called, has been 
known as a variable star since 1670, when Montanari not only 
noted the fact of variation in light but actually observed it at 

* [Kopal, Zdenek; A Study of the Algol System, Ap. Jour. 96 (3), 399-420, 

^Elem^nts of the Third Orbit: P: 1.873 yrs.; e: 0.26; i (provisional): 72°; 

Mass: 1.0 + 0.30. 

Semimajor axis of relative orbit (a t + a,): 36 x 10' km; apparent separa- 
tion at maximum elongation: O'.'IO. — J.T.K.] 



254 THE BINARY STARS 

minimum brightness* on November 8, but the general character 
of its light variation was first established by Goodericke, in 1783. 
He found that the successive minima occurred at intervals of 
about 2 20 49 m , the descent to, and recovery from, minimum 
covering about eight or nine hours, and that for the rest of the 
period the light remained sensibly constant. He explained the 
phenomena by assuming that a dark companion, revolving 
about a common center with Algol, produced a partial eclipse of 
the bright star once in each revolution, but this hypothesis, 
though essentially correct, as we now know, was not generally 
accepted and, in fact, was almost forgotten, until Pickeringf 
revived it in 1880, and Vogel,t in 1889, found the star to be a 
spectroscopic binary, with a period equal to that of its light 
variation. 

The star has been the subject of so many memoirs that it would 
be impossible to give even a full reference list of them here, to 
say nothing of an adequate account of the work they represent 
or of the theories advanced in them. We must limit ourselves 
to a general description of the more significant advances made 
in our knowledge of the system. 

Argelander§ was the first to demonstrate the existence of 
fluctuations in the period between successive light minima, 
and Chandler's H more extensive studies, utilizing all available 
observations from Goodericke's time to 1888 (and in his later 
work to 1897), not only confirmed this conclusion but led him to 
explain them as arising from a long-period inequality which he 
ascribed to the influence of a third body in the system. 

His formula for the period failed, however, to represent later 
epochs of minima, and that has also been the fate of more 
recently derived formulas, including Hellerich's (1919). There 
still remain small periodic and irregular variations not fully 
accounted for by theory. 

Belopolsky's discovery, in 1906, of a variation in the radial 
velocity of the center of mass of the eclipsing system in a period 
of the order of 1.8 years, was the next step in advance. The 
work of Curtiss in 1908, and, more conclusively, that of Schles- 
inger in 1912, established the existence of the third body called 

* See Porro's note, A.N. 127, 41, 1891. 
t Proc. Amer. Acad. 16, 1, 371, 1880. 
tPubl. Potsdam Obs. 7, 111, 1889. 
§ Bonn-Beob. 7, 343, 1869. 
If A.J. 7, 165ff, 1888; 22, 39, 1901. 



SOME BINARY SYSTEMS OF SPECIAL INTEREST 255 

for by this variation. Curtiss gave 1.899 years for the revolution 
period, Schlesinger, 1.874. The latter also found the semi- 
amplitude of the velocity curve to be 9.14 km, and the orbit to 
be nearly circular. The revolution of the eclipsing pair about the 
center of mass in this long-period orbit, produces an oscillation in 
the times of minimum light, which, as Schlesinger points out 
may amount to a displacement of five minutes in either direction. 
Meanwhile, Stebbins had perfected his selenium photometer, 
so that in the years 1909 and 1910, he was able to measure the 
brightness of stars to the third magnitude with far greater 
accuracy than had been possible by visual or photographic methods. 
Applying it to the study of Algol, he discovered the existence of 
a secondary minimum in its light curve, with a depth of but 
0.06 magnitude and showed, moreover, that the light varied 
continuously between minima. Ten years later he repeated the 
investigations with the far more sensitive photoelectric-cell 
photometer. The new light curve was, naturally, more accurate 
than the earlier one, but confirmed the secondary minimum and 
the continuous variation in the light. It also showed an effect 
resulting from the ellipsoidal shape of the components. The orbit 
of the eclipsing system was shown to be practically circular. 

The rotation of a star upon its axis has the effect of broadening 
the lines of its spectrum unless the axis is directed toward us 
since one limb is receding, the other approaching us relatively 
to the motion of the star's center. This effect, which has received 
special attention from O. Struve in his studies of the emission 
lines in Class B stars, is particularly pronounced in the case of 
eclipsing binary stars, since the receding limb of the component 
entering eclipse is the visible one, whereas the approaching 
limb is the visible one on emerging from eclipse. Rossiter 
studied this effect in j8 Lyrae, and McLaughlin, in Algol, in 
1923 and 1924. In both cases it was found to be well marked, 
the residuals before minimum light in Algol, with which we are 
here concerned, all being positive and those after minimum 
negative, as theory requires. The observed range of the effect 
was 35 km. Using Stebbins's values for the relative dimensions 
of the system and for the inclination of the orbit to compute the 
range, McLaughlin found it necessary to assume the brighter 
star to be five times as massive as the fainter one to secure 
agreement with the observed value. On this assumption, with 
the observed range of the rotation effect, it becomes possible to 



256 THE BINARY STARS 

compute the absolute dimensions in the system, the mass and 
density of each component, and also the parallax. 

Combining McLaughlin's spectrographs results with Steb- 
bins's photometric ones we have the following data for the 
eclipsing system: 

Period (Hellerich's value) 2^867301 

Duration of principal minimum 9* 66 

Inclination of orbit (cos i = 0.142) 81? 8 

Distance between centers (ab + a/) 10,522,000 km 

Radius of bright body 3.12© 

Radius of faint body 3 . 68 O 

Mass of bright body 4 . 72 O 

Mass of faint body . 95 O 

Density of bright body 0. 16 O 

Density of faint body . 02 Q 

The corresponding dynamical parallax of Algol is +0-'031. 

It is specially to be noted, however, that nothing is yet known 
of the spectrum of the third star or of any effect it may have 
upon the photometric measures of the eclipsing pair.* 

McLaughlin, in his closing paragraphs, refers to the difficulty 
of obtaining a reliable trigonometric parallax for Algol, because 
of its revolution in an orbit about two-thirds as large as the 
Earth's orbit in a period of less than two years. This is a point 
to which Wright called attention as long ago as 1904, in his 
note on the parallax of a Centauri and which he elaborated more 
fully in 1921 in his notef "On Spectroscopic Binaries and the 
Determination of Parallax." It is, obviously, one to be specially 
noted by parallax observers. 

References 

It is impossible^ and, happily, unnecessary to list here the many papers 
that have been consulted in the preparation of this chapter. Footnote 
references have been given to a number of the more important ones, and the 
extensive catalogues of double stars, of spectroscopic binary stars, and of 
variable stars will help any reader who wishes to look up more fully any of 
the systems that have been described. 

* It should be noted that M. A. Danjon in his photometric study of 
Algol (Annates, Strasbourg Obs. 2, 148, 1928) reaches conclusions differing 
in many particulars from those described above. These conclusions, 
however, await confirmation. 

jPubl. A.S.P.Z3,47, 1921. 

X The references to the literature on Algol alone would fill four quarto 
pages of fine print! It is hardly necessary to add that not all of these 
papers were read in preparing my note. 



CHAPTER X 

STATISTICAL DATA RELATING TO THE VISUAL BINARY 
STARS IN THE NORTHERN SKY 

The visual binary stars for which orbits have been computed 
or in which the observed arc suffices for the computation of 
dynamical parallaxes may be utilized, as has been shown in 
Chap. VIII, in the study of the relations between the orbit 
elements and for the investigation of stellar mass. It is obvious, 
however, that even if they were not selected stars, they would 
not afford an adequate basis for a study of the number of the 
visual binaries, of their distribution (apparent or real), or of 
the absolute magnitudes of their components. The data required 
for a thoroughly satisfactory investigation of these and similar 
problems would include a complete enumeration of all binary 
systems to a definite limiting stellar magnitude, and measures 
(a) of the angular distances of all pairs for a given epoch, (6) 
of the photometric magnitudes and spectral classes of both 
components, and (c) of the parallaxes of the systems. 

The first step in an approximation to this ideal material is 
to make a survey of all stars in the sky to a definite magnitude 
with sufficiently powerful telescopes used under good observing 
conditions. All previously known pairs must be noted and all 
new pairs falling within predetermined limits of magnitude and 
angular separation identified and measured. The Lick Observa- 
tory survey and the surveys of the southern sky which are still 
in progress at Johannesburg and Bloemfontein, all described in 
my historical sketch, were undertaken with the definite purpose 
of providing the basis for such statistical studies. 

The Lick Observatory survey was completed in 1915, and the 
results obtained for the sky area north of the equator will now be 
presented. The data consist of all known double stars as 
bright as 9.0 BD magnitude* which fall within the distance 
limits set by the following working definition of a double star 

* BD magnitudes were taken because photometric magnitudes to the 
limit 9.0 were not available. 

257 



258 THE BINARY STARS 

proposed by me in 1911: 

(1) Two stars shall be considered to constitute a double star when 
the apparent distance between them falls within the following limits: 

1" if the combined magnitude of the components is fainter than 11.0 
3" if the combined magnitude of the components is fainter than 

9.0 BD. 
5" if the combined magnitude of the components lies between 6.0 
and 9.0 BD. 
10" if the combined magnitude of the components lies between 4.0 

and 6.0 BD. 
20" if the combined magnitude of the components lies between 2.0 

and 4.0 BD. 
40" if the combined magnitude of the components is brighter than 
2.0 BD. 

(2) Pairs which exceed these limits shall be entitled to the name 
double star only when it has been shown (a) that orbital motion exists; 
(6) that the two components have a well defined common proper motion, 
or proper motions of the 61 Cygni type; (c) that the parallax is decidedly 
greater than the average for stars of corresponding magnitude, t 

In all, there are 5,400 pairs, more than half of which were 
discovered in the course of the survey (766 by Hussey, 2,057 by 
* [See pages 35 and 268. Aitken's working definition (1911) is: 
log P = 2.5 - 0.2m 
which gives 5" for 9.0 magnitude. 
Also, see ADS, p. IX, separation limits for ADS 
log p = 2.8 - 0.2m 



pp. m 


P 


App. m 


P 


1.0 


400" 


7.0 


25" 


2.0 


250" 


8.0 


16" 


3.0 


160" 


9.0 


10" 


4.0 


100" 


10.0 


6" 


5.0 


63" 


11.0 


4" 


6.0 


40" 


12.0 


275— J.T.K.] 



t The definition, with correspondence relating to it, will be found in the 
Astronomische Nachrichten (188, 281, 1911). Comstock and E. C. Pickering 
there suggest limits based upon the apparent magnitude, the former using 
the formula s = c($j) m , the latter, the formula, log s = c — 0.2m, where s 
is the distance in seconds of arc between the components, m, the apparent 
magnitude, and c an arbitrary constant. If the values of c in the two for- 
mulae are so chosen as to give the limit 5-'0 for stars of magnitude 6.0, the 
formulae will give the limits 0-'75 and 1 "25, respectively, for stars of 9.0 mag- 
nitude. From the theoretical point of view either formula gives more 
logical limits than the ones in my definition, but there were practical con- 
siderations, fully stated in the article referred to, which led to the adoption 
of the latter. 



DATA RELATING TO STARS IN THE NORTHERN SKY 259 

Aitken). A given system is counted only once though it may 
have three, or even four or more, components. In the multiple 
systems the closer pair is, in general, the one counted, but in a 
few cases in which the close pair is very faint, the principal 
bright pair is taken. 

The first question to consider is whether the data are homo- 
geneous, for it is obvious that they can make no claim to be 
exhaustive. However carefully an observer may work, some 
pairs which he might discover with a given telescope will surely 
escape him. His eye may be fatigued, unnoticed haze or 
momentary bad seeing may blur out a faint companion star, 
or it may chance that at the date of examination the two com- 
ponents are so nearly in conjunction as to be below the resolv- 
ing power of the telescope. The number of known double 
stars can only be regarded as the lower limit to the number 
which might be discovered. Homogeneity was earnestly 
sought for, care being taken to work only when in good phys- 
ical condition and when the seeing was good, the practical test 
being the power to recognize very close and difficult pairs at 
a glance. But variations in the conditions are inevitable when 
the working program requires years for its execution and doubt- 
less such variations have affected the present results. Careful 
comparison, however, shows no discernible difference in the 
thoroughness of the work done at different seasons of the year 
or in different parts of the sky, and it may fairly be said that 
the results of the survey represent the capacity of the com- 
bination of telescope and observer under average good atmos- 
pheric conditions at Mount Hamilton. If the work had all 
been done with the 36-in. refractor the resulting data might 
be considered quite homogeneous. Unfortunately, a con- 
siderable part of it, including practically the entire area north 
of +60° declination, was done with the 12-in. telescope, and it 
becomes necessary to consider the relative efficiency of the 
two instruments. 

I have applied two tests: first, the comparison of the most 
difficult pairs discovered with each instrument; second, the 
reexamination with the 36-in. of some 1,200 stars previously 
examined with the 12-in. telescope. I find that, under the 
usual observing conditions, a pair with nearly equal components 
separated by only 0"15, or a companion star as faint as 14.5 mag- 
nitude and not less than 1"5 from its primary is practically 



260 



THE BINARY STARS 



certain of detection* with the 36-in.; with the 12-in., the cor- 
responding limits in the two cases are 0"25 and 13 to 13.5 mag- 
nitude. Twelve new double stars were added by the reexamina- 
tion of the 1,200 stars. From these tests, taking into account 
the proportion of the whole work done with the 12-in. telescope, 
I conclude that about 250 pairs would have been added if the 
entire northern sky had been surveyed with the 36-in. Since 
all or nearly all of the brighter stars had been examined repeatedly 
by other observers using powerful telescopes, it is fair to assume 
that comparatively few of these undiscovered pairs are brighter 
than 7.0 BD magnitude. 

According to Seeliger's count of the BD stars there are 100,979 
as bright as 9.0 magnitude in the northern hemisphere. Of 
these, 5,400, or 1 in 18.7 on the average, have actually been 
found to be double within the limits set above. If we add 
only 200 pairs, the ratio becomes 1 : 18.03. A definite answer is 

Table 1. — The Distribution of Double Stars in Right Ascension and 

Declination 



R.A. 
Decl. 


0>- 

l h 


2 h - 
3 h 


4 h - 
5 h 


6 h - 

7 h 


8 h - 
9" 


ltf 1 - 
ll h 


12 h - 
13 h 


14 h - 
15 h 


16 h - 
17 h 


18 h - 
ltf 1 


20 h - 
21 h 


22 h - 
23 h 


0°-9° 


6.3 
5.4 
5.2 
6.0 
6.7 
6.2 
4.6 


6.3 
5.4 
5.6 
5.5 
5.0 
6.2 
5.0 


7.4 
6.4 
5.9 
5.9 


6.0 


6.0 
5.5 
4.8 
4.3 
4.5 
4.8 
4.3 

3.1 

ll h ) 


5.9 
5.5 
6.4 

4.8 
6.2 
3.6 
4.4 
2.6 
3.3 


5.5 

4.8 
5.2 
5.4 
3.2 

4.4 

2.6 

2.8 
(12 h - 


6.4 
6.1 

4.8 
4.1 
5.6 
6.0 
4.1 
2.2 
-17 h ) 


5.4 
4.5 
5.4 
5.0 
5.5 
4.6 
3.6 
3.4 
3.4 


6.0 
6.0 
6.2 


5.3 
5.4 
4.9 
5.2 
5.2 


4.9 


10-19 
20-29 


6.2 
5.1 
4.4 
4.5 
5.2 
2.9 
3.5 
(6 h - 


5.2 

4.2 


30-39 


5.6 
4.2 
4.4 
4.1 
4.9 
(18 h - 


4.8 


40-49 


5.0 
8.0 
7.1 
4.2 
3.7 


4.9 


50-59 
60-69 


6.8 
5.3 
5.0 

-23 h ) 


4.9 
6.8 


70-79 
80-89 


5.S 

(0 h - 


► 3.8 

-5 h ) 


6.5 
3.9 



The figures give the percentages of double stars among stars to 9.0 BD magnitude; the 
average percentage for the whole northern sky is 5.35. 

thus given to my first question: At least one in every 18, on the 
average, of the stars in the northern half of the sky which are as 
bright as 9.0 BD magnitude is a close double star visible with the 
36-t'n. refractor.f There is no reason to doubt that the ratio is 

* Unless the primary is brighter than 7.0 magnitude. 

t [Research and observation since 1935 lead to the conclusion that at least 
half the stellar population consists, not of single stars like our sun, but of 
members of systems; that is, practically one out of every two stars is a binary 
or multiple system. Also, in spite of the high frequency of spectroscopic 
binaries among bright stars, W. H. van den Bos assures us that in space the 
spectroscopic binary is a rare exception, the visual binary the rule.— J.T.K.] 



DATA RELATING TO STARS IN THE NORTHERN SKY 261 

equally high in the southern half of the sky. In fact, preliminary 
counts made by van den Bos lead him to estimate the ratio in 
that hemisphere as high as one in 17. 

Table 1 exhibits the distribution of the 5,400 double stars in 
right ascension and declination as compared with the distribu- 
tion of the BD stars to 9.0 magnitude, the figures giving the 
percentage of double stars in each area. There are obvious 
irregularities in the table but no evidence of systematic differ- 
ences that can be regarded as seasonal effects. The percentages 
are as high in the sky areas surveyed in winter as in those sur- 
veyed in summer. There is a falling off in the percentage in 
the high declinations, especially in the regions well removed 
from the Milky Way, which is doubtless due in part to the 
fact that the area north of 60° was almost entirely surveyed 
with the 12-in. telescope. The broken line in the table represents 
very roughly the position of the central line of the Milky Way, 
and it will be noted that the percentages near this line are, in 
general, above the average. 

The distribution with respect to the plane of the Milky Way 
is more clearly brought out when the stars are tabulated accord- 
ing to galactic latitude. This has been done in Tables 2 and 3, 
in the former of which the stars are divided into classes accord- 
ing to magnitude and the latitudes into zones each 20° wide, 
Table 2. — The Distribution of Double Stabs by Magnitude Classes 
and Zones op Galactic Latitude 



r/^^v^ Mag. 


to 6.5 


6.6-7.0 


7.1-7.5 


7.6-8.0 


8.1-8.5 


8.6-9.0 


Total 


Zone ^n. 
















I 


19 


13 


14 


29 


40 


84 


199 


II 


43 


28 


50 


68 


114 


193 


496 


III 


60 


43 


56 


79 


148 


254 


640 


IV 


96 


54 


81 


132 


232 


401 


996 


V 


121 


88 


133 


249 


376 


653 


1,620 


VI 


84 


51 


81 


134 


221 


395 


966 


VII 


28 


23 


18 


54 


90 


154 


367 


VIII 


7 


6 


5 


12 


31 


55 


116 


Total 


458 


306 


438 


757 


1,252 


2,189 


5,400 



beginning at the north galactic pole. Zone V therefore includes 
the area from +10° to —10° galactic latitude, and zone IX, 
which ends at the south galactic pole and lies entirely below 
the equator, is not represented. As was to be expected, the 
numbers in every column of this table are largest in zone V 



262 



THE BINARY STARS 



and fall to minima in zones I and VIII. The question is whether 
this condensation toward the Milky Way is greater than that 
of all the stars. Table 3 provides the answer. Since the 
zones are not of equal area, and since only the first one lies 
wholly in the northern hemisphere, the fairest comparison 
is that afforded by the relative densities per square degree of 
double stars and of all stars of the corresponding magnitudes. 
The double star densities were determined by dividing the 
figures in Table 2 by the number of square degrees in each 
zone area; the figures were then reduced to a common standard 
by making the density in each column unity in zone V, the 
Milky Way zone. Seeliger has published corresponding data 
Table 3. — Density op Double Stabs by Magnitude Classes and 
Galactic Latitude Compared with the Density op BD Stabs to 
9.0 Magnitude (Afteb Seeligeb) 



n. Mag. 


to 6.5 


6.6-7.0 


7.1-7.5 


7.6-8.0 


8.1-8.6 


8.6-9.0 


Zone >v 


BD 


D.S. 


BD 


D.S. 


BD 


D.S. 


BD 


D.S. 


BD 


D.S. 


BD 


D.S. 


I 

II 

III 

IV 

V 

VI 

VII 

VIII 


0.551 
0.572 
0.639 
0.790 
1.000 
0.912 
0.572 
0.428 


0.395 
0.456 
0.480 
0.789 
1.000 
0.822 
0.395 
0.307 


0.431 
0.445 
0.554 
0.689 
1.000 
0.787 
0.427 
0.315 


0.374 
0.410 
0.474 
0.610 
1.000 
0.687 
0.446 
0.361 


0.518 
0.497 
0.599 
0.765 
1.000 
0.842 
0.467 
0.352 


0.266 
0.484 
0.407 
0.606 
1.000 
0.721 
0.231 
0.199 


0.404 
0.424 
0.509 
0.730 
1.000 
0.772 
0.480 
0.373 


0.304 
0.351 
0.307 
0.529 
1.000 
0.639 
0.370 
0.255 


0.419 
0.441 
0.512 
0.720 
1.000 
0.799 
0.521 
0.462 


0.261 
0.390 
0.380 
0.614 
1.000 
0.694 
0.407 
0.435 


0.382 
0.404 
0.484 
0.728 
1.000 
0.789 
0.527 
0.527 


0.325 
0.380 
0.377 
0.613 
1.000 
0.716 
0.401 
0.445 



for all of the BD stars and the two sets of values are entered 
in Table 3 in the columns D.S. and BD, respectively. It is 
clear that the density curves of double stars rise to sharper 
maxima in the zone V than the corresponding curves of stars 
in genera] do. 

This fact is exhibited in a more striking manner if we tabulate, 
as in Table 4, the percentages of double stars in five areas, the 
Milky Way zone, the 20° zone on either side of it and the areas 
north of +30° and south of —30° galactic latitude. 

The increased percentage in zone V must be accepted as 
real. Table 3 shows that stars of all magnitude classes par- 
ticipate in it, and an examination of my charts leads to the 
conclusion that it cannot be an observing effect, for some areas 
of all galactic latitudes were examined in summer, others in 
winter; the area north of +60° declination, examined almost 
exclusively with the 12-in., extends from —3° to +27° galactic 



DATA RELATING TO STARS IN THE NORTHERN SKY 263 

latitude and the areas of high galactic latitude, both north and 
south, were examined mainly with the 36-in. refractor. It 
appears, therefore, that among stars as bright as 9.0 BD magni- 
tude close visual double stars are relatively more numerous in 

low than in high galactic latitudes. 

Table 4. — Percentages of Double Stars 



Galactic latitude 


BD stars to 9.0 


Double stars 


Percentage of 
double stars 


+90° to +30° 
+30 +10 
+10 -10 
-10 -30 
-30 -70 


26,948 
19,355 
26,477 
17,831 
10,368 


1,335 
996 

1,620 
966 

483 


4.95 
5.15 
6.13 
5.13 
4.66 



This apparent concentration of double stars toward the galactic 
plane is certainly to be explained, in part, by the far greater 
extension of the stellar system in that plane than in the direction 
perpendicular to it. Possibly this is the full explanation, 
perhaps the observed increase in double star density is entirely 
a perspective effect; but in that event it would seem that in 
zone V, the galactic zone, we might expect a relatively higher 
percentage of very close pairs than of pairs of greater separation. 
Table 5, however, in which the 5,400 pairs are grouped according 
to galactic latitude and angular separation, shows that the 
percentage increase toward zone V is substantially the same in 
all the angular distance categories up to 5" 00. 

Let us consider next the relation between the angular sep- 
aration and magnitude. This is shown in Table 6 where the 
pairs are arranged with these qualities as arguments. The 

Table 5. — The Distribution of Double Stars in Galactic Latitude 
by Distance Classes 



\Dist. 


o'.'oo- 


0751- 


1701- 


1751- 


2"01- 


3701- 


4701- 


5701 


\ 


0750 


1700 


1750 


2700 


3700 


4700 


5700 


and over 


Zone\ 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


No. 


% 


I 


41 


11 


31 


10 


25 


14 


24 


16 


39 


17 


15 


9 


14 


10 


10 


20 


II 


101 


26 


92 


30 


56 


30 


56 


38 


61 


26 


65 


40 


48 


33 


17 


34 


III 


139 


36 


99 


32 


90 


49 


65 


44 


101 


43 


61 


37 


53 


36 


32 


64 


IV 


225 


58 


175 


57 


131 


71 


104 


70 


133 


57 


112 


68 


74 


51 


42 


84 


V 


388 


100 


306 


100 


184 


100 


149 


100 


233 


100 


164 


100 


146 


100 


50 


100 


VI 


247 


64 


152 


50 


119 


65 


105 


70 


144 


62 


101 


62 


72 


49 


26 


52 


VII 


82 


21 


77 


25 


52 


28 


36 


24 


42 


18 


32 


20 


29 


20 


17 


34 


VIII 


33 


8 


18 


6 


14 


8 


12 


8 


12 


5 


9 


6 


12 


8 


6 


12 



264 



THE BINARY STARS 



Table 6. — The Distribution of Double Stabs bt Angulab 
Distance and Magnitude 



\Dist. 


to 


0-'51- 


o-'oo- 


l"01- 


2"01- 


3-'01- 


4''01- 


5"01 
and 
over 


Mag.\ 


0"50 


l-'OO 


l-'OO 


2"00 


3-'00 


4"00 


5"00 


g 6.5 


75 


63 


138 


83 


62 


41 


31 


99 


6.6-7.0 


82 


52 


134 


59 


42 


40 


21 


14 


7.1-7.5 


103 


67 


170 


99 


64 


48 


31 


29 


7.6-8.0 


178 


132 


310 


164 


107 


85 


63 


26 


8.1-8.5 


310 


223 


533 


285 


173 


128 


111 


21 


8.6-9.0 


508 


413 


921 


532 


317 


217 


191 


11 


Totals 


1,256 


950 


2,206 


1,222 


765 


559 


448 


200 



Percentages 



^ 6.5 
6.6-7.0 
7.1-7.5 
7.6-8.0 
8.1-8.5 
8.6-9.0 

Totals 



39 
45 
41 
42 
43 
42 



42 



23 
20 
24 
22 
23 
25 



23 



17 
14 
16 
15 
14 
14 



15 



12 
14 
12 
12 
11 
10 



11 



sums of the numbers in the first two columns of the table are 
entered in the third, thence the numbers are given for uniform 
steps in angular distance to the final column. Every line of 
columns three to seven exhibits a marked increase in the number 
of pairs as the angular distance diminishes. 

This is still more apparent when the figures in these five col- 
umns are expressed as percentages of the total number of pairs 
under 5"0 separation in each magnitude class. If we may 
assume that the stars of a given magnitude class, for example, 
from 8.6 to 9.0, are, on the average, at the same distance from 
us, then this observed increase in the number of pairs as the 
angular distance diminishes is not a mere perspective effect 
but represents a real increase in the number of pairs as the angular 
distance diminishes. 

The table also indicates that the Lick Observatory survey 
was as thorough for the fainter stars (to 9.0 BD magnitude) 
as for the brighter ones, for the percentages in each column 
in the lower division of the table are not far from uniform and 
there is a fair agreement between the ratios of the figures in each 
line of the first two columns of the upper half of the table. This 
was to be expected, for the pairs most likely to be missed in a 



DATA RELATING TO STARS IN THE NORTHERN SKY 265 



survey are those in which the difference in magnitude of the 
two components is large and bright pairs of that character are 
at least as difficult objects as pairs in which the primary is 
of 9.0 magnitude.* When the two components are of equal 
brightness, on the other hand, a 9.0 magnitude pair is but little 
more difficult than one of 6.0 magnitude, unless the angular 
distance is below 0"15; in fact, I find that 123 of the 379 pairs 
with angular distance less than, or equal to, 0"25, and 385 of 
the 877 pairs with angular distances between 0"26 and 0"50, 
discovered in that survey are of the BD magnitude class 8.6-9.0. 
These statements are of significance also in connection with the 
figures entered in Table 7, which shows the percentage of double 
stars of each BD magnitude class. 

Table 7. — Percentage of Double Stabs by Magnitude Classes 



Magnitude 



to 6.5 
6.6-7.0 
7.1-7.5 
7.6-8.0 
8.1-8.5 
8.6-9.0 



BD stars 



4,120 

3,887 

6,054 

11,168 

22,898 

52,852 



Double stars 



458 
306 
438 
758 
1,251 
2,189 



Percentage of 
double stars 



11.1 
7.9 
7.2 
6.8 
5.5 
4.1 



The drop in percentage as we pass from the brighter to the 
fainter magnitudes cannot be attributed to lack of completeness 
in the data, for it follows from what has been said above, that the 
pairs overlooked in the survey are quite as likely to belong to 
the brighter magnitude classes as to the fainter ones; but we must 
not forget that this table, as well as the preceding ones, is based 
upon the apparent magnitudes only; whether the observed 
progression holds also for the absolute magnitudes we shall 
not know until we have more knowledge of the parallaxes of 
the stars involved. The very high percentage of spectroscopic 
binaries, among stars as bright as or brighter than 5.5 apparent 
magnitude, and particularly among stars of classes A and B, 
may possibly be significant in this connection, though we cannot 
as yet say definitely that the percentage is not as great among the 
fainter stars. 

It will be of interest to inquire whether the results for the 
relative numbers and apparent distribution of the visual binary 

* This holds true unless Am exceeds six magnitudes. 



266 



THE BINARY STARS 



stars given by the survey just reviewed find any support from 
the data given in the New General Catalogue of Double Stars 
within 120° of the North Pole, or from other available data. 
In 1932, I made a single count of the double stars in the new 
catalogue which are included within the limits of the curve 
defined by the formula log p = 2.5 — 0.2m. This formula 
gives 5''0 as the angular separation corresponding to a pair of 
9.0 magnitude. I found 12,708 such pairs. Of these, 4,761 pairs 
are fainter than 9.0 magnitude and must be omitted in our 
present inquiry because no adequate double star survey has 
yet been made of the stars below 9.0. Further, it must be said 
that the survey is not complete even to 9.0 for the entire sky 
down to —30° declination. Making a very generous estimate 
of the number of such stars actually examined, I again find that 
at least one in every 18 is a close double star within the resolving 
power of good modern telescopes. 

Table 8 shows the distribution of these 7,947 pairs by magni- 
tude and angular separation, the data being given in more 
condensed form than was adopted in the earlier tables to minimize 
the effect of errors in the rapid count of the different categories. 

Table 8 



^\. Ang. sep. 


<0''51 


0-'51- 
l''00 


l"01- 
2''00 


2''01- 
5"00 


5-'01- 
Curve 


Total 


< 6.00 
6.01-7.00 
7.01-8.00 
8.01-9.00 


66 
136 
305 
879 


69 

98 

244 

793 


63 

134 

346 

1,035 


156 

214 

522 

1,564 


317 
302 
341 
363 


671 

884 

1,758 

4,634 


Totals 
Percentages 


1,386 

17.4 


1,204 
15.2 


1,578 
19.9 


2,456 
30.9 


1,323 
16.6 


7,947 
100.0 



The number of pairs under 0'-'51 in angular distance exceeds 
the number with distances between 0"51 and l'-'OO; the number 
under l'-'OO is about one-third, the number wider than 5"0 is 
but one-sixth of the whole. These figures support the earlier 
finding that the number of double stars increases as the angular 
distance diminishes. 

The fact that the tables presented above are based upon the 
apparent magnitudes, the observers' estimates (often very 



DATA RELATING TO STARS IN THE NORTHERN SKY 267 

inexact) of the difference in magnitude of the two components 
of a pair, and upon measures of the angular distances made 
at many different epochs and with telescopes both large and 
small, is to be emphasized. If accurate photometric measures 
of magnitude (and of Aw) were available, and fairly reliable 
measures of all angular distances, it would be desirable to set up 
tables with the arguments log distance, m, and Aw. Such 
tables would give a far better picture of the actual distribution 
of the double stars. 

It is well known that four of the ten stars nearest to the Sun 
are visual binaries: a Centauri, Sirius, Procyon, and 61 Cygni. 
This, no doubt, is an exceptional percentage, but it is of interest 
to quote some figures sent me by van Maanen who has recently 
made a study of the nearest stars.* Within a distance of 
5 parsecs (approximately 16 light years), he finds that 8 double 
star systems (with 18 components) and 21 single stars (including 
the Sun) are known; in the volume of space between the limits 
5 and 10 parsecs, 16 double or multiple systems (38 components; 
and 62 single stars, and in the volume between 10 and 20 parsecs, 
91 double or multiple systems (197 components) and 281 single 
stars. In all, 617 individual stars (including the Sun) are known 
to us in the volume of space within a radius 'of 20 parsecs. Of 
these, 364 are single stars; the remaining 253 are components of 
115 double or multiple systems. Counting each such system 
as a unit, then, one star in four is a visual double or multiple. 
Van Maanen proceeds to show that there are probably about 
2,000 individual stars in this volume of space that have not 
yet been recognized as inhabitants. The chances are that the 
percentage of components of double star systems is not so great 
among these fainter stars as among those already known. 
Counts like these cannot be compared directly with the results 
given by our double star survey, but they nevertheless indicate 
that our estimate of the frequency of double stars is a very 
conservative one. 

The spectral classes of the stars have been ignored in the 
preceding tables. It will be of interest now to take them into 
account, as far as possible. In 1917, through the courtesy of 
the Director, the late E. C. Pickering, and of Dr. Annie J. Cannon, 
of the Harvard College Observatory, I had the privilege of 
comparing my list of 5,400 double stars with the great card 

* Publ. A.S.P. 46, 247, 1933. 



268 



THE BINARY STARS 



catalogue of stellar spectra later published as the Henry Draper 
Catalogue. This comparison provided the spectral classification 
of 3,919 of the 5,400 pairs. Of the remaining 1,481, only 15 
are as bright as 8.0 (BD), 218 lie between 8.1 and 8.5, and 
1,248 between 8.6 and 9.0. The published volumes of the 
Henry Draper Catalogue were available when I prepared the 
copy for my New General Catalogue of Double Stars within 120° 
of the North Pole (briefly the ADS), and I utilized them to enter 
all known spectra of the double stars I catalogued. I have made 
a single count of these, taking the spectrum of the primary star 
only, in cases where the spectra of additional components are 
entered, and find the total to be 9,190. 

The pairs in the ADS, include all double stars to —30° declina- 
tion, falling within the very generous distance limits set by the 
curve log p = 2.8 — 0.2m, which gives the angular distance 
10"0 for a 9.0 (BD magnitude) pair. 

Table 9 shows the distribution according to spectral class of 
the 9,190 ADS pairs with known spectra, of the 3,919 pairs in 
the northern half of the sky which fall within the much narrower 
limits set by my "working definition," and of the 222,570 stars 
in the Henry Draper Catalogue, as given in H.C.O. Circular 226. 

Table 9 
Numbers 



Visual pairs in n. hem. , 
Visual pairs in ADS . . . 
Stars in HDC 



Ratio ADS: HDC. 



B 


A 


F 


G 


K 


M 


157 


1,251 


532 


1,093 


837 


49 


268 


2,910 


1,220 


2,461 


2,126 


205 


3,567 


64,259 


21 , 120 


46,552 


73,208 


13,864 


1:13 


1:22 


1:17 


1:19 


1:34 


1:68 



All 

3,919 

9,190 

222,570 



Percentages 



Visual pairs in n. hem. 
Visual pairs in ADS . . 
Stars in HDC 



4.0 


31.9 


13.6 


27.9 


21.4 


1.2 


2.9 


31.7 


13.3 


26.8 


23.1 


2 2 


1.6 


28.9 


9.5 


20.9 


32.9 


6.2 



100.0 
100.0 
100.0 



Table 10, in similar manner, shows the distribution of the 
spectroscopic binary stars in Moore's Third Catalogue (Table I), 
omitting the Cepheid variables and other stars whose variation 
in radial velocity does not satisfy Otto Struve's criterion. Since 
these binaries are practically all bright stars — nearly all brighter 



DATA RELATING TO STARS IN THE NORTHERN SKY 269 



than 5.5 visual magnitude — their distribution in spectral class 
is compared with that of the HDC stars as bright as 6.25 magni- 
tude (H.C.O. Circ. 226). 



Table 10 
Numbers 





B 


A 


F 


G 


K 


M 


All 


Spec. Bin. 

HDC Stars to 6.25 

Ratio Spec. Bin. to HDC 


227 
719 

1:3 


360 
2,018 

1:6 


104 
680 

1:7 


115 
656 

1:6 


149 
1,984 

1:13 


26 
538 

1:21 


981 
6,595 



Percentages 



Spec. Bin. 

HDC Stars to 6.25 



23.1 
10.9 



36.7 
30.6 



10.6 
10.3 



11.7 
9.9 



15.2 
30.1 



2.7 
8.2 



100.0 
100.0 



The upper part of each table records the actual numbers of 
stars counted, the lower part their percentage distribution among 
the spectral classes. It is evident that the count of the 3,919 
visual parts made in 1917 gave a good representation of the 
apparent distribution of the visual binary stars among the 
different spectral classes, for the count of the 9,190 pairs exhibits 
very closely the same percentage distribution. 

It appears from the frequency ratios in Table 9 that visual 
binary stars are relatively more numerous among stars of 
classes B, F, and G, than among the stars of the other three 
classes, the small ratios for classes K and M being specially 
striking. From Table 10 it is equally clear that among the stars 
as bright as 6.25 magnitude, spectroscopic binaries are most 
numerous among stars of Class B and least numerous among 
stars of Class M. The strong contrast between the frequency 
ratios for classes B and M in both tables is perhaps their most 
striking feature. It may be of interest to add that my count 
shows that the visual binaries of classes B and A in the ADS 
are strongly concentrated toward the galactic plane, whereas 
those of classes F and M are quite uniformly distributed over the 
sky, and those of classes G and K are more frequent among stars 
of high than of low galactic latitude. 

Such general statistics are of interest and have also a certain 
degree of significance, as have also the statistical relations 



270 THE BINARY STARS 

between the magnitudes, spectral classes and colors of the 
components of the visual binary stars. The relation between 
the colors of the components of double stars and their difference 
in magnitude was recognized by Struve and every observer 
since his time has noted that fact that when the two components 
are about equally bright they are almost without exception of 
the same or nearly the same color, and that the color contrast 
increases with the difference in magnitude of the components. 
Professor Louis Bell argued that this is a subjective effect 
since the fainter star is generally the bluer one. Doubtless this 
subjective effect is often present but it is by no means the sole 
cause. There are real and very striking differences in the 
spectral classes of the components of double stars and these 
are definitely correlated to the color differences and also to the 
differences in magnitude. The absolute magnitude of the 
primary also enters as a factor. Thus Lau,* in two papers 
written as early as 1917 and 1918, respectively, found that the 
companions to giant stars are bluer, the companions to dwarf 
stars redder than their primaries. 

Several writers have investigated the relations between 
magnitude and spectral class, in the components of double stars, 
among them Dr. F. C. Leonard, f His thorough analysis of 
the data for 238 pairs clearly showed (1) that when the two 
components are of equal or nearly equal magnitude, they differ 
little in spectral class, except in the case of a few giant stars 
like 7 Circini; (2) that when the primary is a giant of spectral 
class F0 or later, the companion belongs to an earlier spectral 
class; and (3) that when the primary is a dwarf star, or a giant 
star of spectral class earlier than F0, that is, when the primary is a 
star of the main sequence, the companion belongs to a later 
spectral class. These results prove that the color relationships 
observed by Lau correspond to actual differences of spectral 
class. Since they are also in harmony with the mass-luminosity 
relationship, they indicate that the components of double stars 
are normal stars, having the same properties as ordinary single 
stars of corresponding mass and magnitude. 



* A.N. 205, 29, 1917; 208, 179, 1918. 

t Leonard, Lick Obs. Bull. 10, 169, 1923. See also Peter Doig's paper 
(Mon. Not. 82, 372, 1922) and G. Shajn's (Bull. Poulkova Obs. 10, 276, 
1925). Their conclusions are similar to Leonard's. 



DATA RELATING TO STARS IN THE NORTHERN SKY 271 

As has already been noted, one of the weakest points in these 
statistical investigations aside from the unavoidable incom- 
pleteness of the data, is that the apparent magnitudes of the 
components of double stars are so unreliable. What is needed 
is the accurate determination of Am (and therefore of difference 
in absolute magnitude) of all pairs to a given magnitude limit 
(for the primary star). Some years ago, Dr. G. P. Kuiper, 
with the aid and encouragement of Prof. Ejnar Hertzsprung, 
began to make such a determination, using suitable wire gratings 
over the 10-in. refractor at Leiden, to reduce the first-order 
spectra of the primary stars to a brightness within one-half a 
magnitude of the image of the secondary star. He is following 
out this program at the Lick Observatory at the present time, 
with the object of including in his observations every star as 
bright as 6.5 visual magnitude on the Harvard scale, that has a 
companion within 30". He is at the same time determining the 
spectral classes of the components, using a slitless spectrograph 
attached to the Crossley reflector. When completed, this 
program will afford a sound observational basis for a study 
of the relationships between Aw and the difference of spectral 
class in these bright pairs. 

The various results as to the number and distribution of the 
binary stars given above are all, as I have said, of interest and of 
significance, though they rest, admittedly, upon data neither 
homogeneous nor complete. Some will be confirmed when 
additional data become available, others may have to be aban- 
doned. One of the most significant findings is perhaps, that the 
components of the binary stars are normal stars, resembling in 
all essentials the single stars of corresponding magnitude and 
spectral class. 

Among other investigations that lead to this same conclusion 
we may refer to Oort's* "A Comparison of the Average Velocity 
of Binaries with That of Single Stars," in which he finds no 
indication of any difference between the two; and Wallinquist'sf 
"The Solar Motion as Derived from the Radial Velocities of the 
Visual Binary Stars." His value for the position of the solar 
apex, derived from a discussion of 536 systems, mainly with 
angular distance between components under 5", agrees closely 
with the generally accepted value. 

* J. H. Oort, A.J. 35, 141, 1924. 

t A. Wallinquist, Ann. Bosscha Sterremoacht 4, 21, 1929. 



272 THE BINARY STARS 



References 

Many papers have been written in the past fifteen years on questions 
related to those discussed in this chapter. In addition to those already 
referred to in the footnotes, a few representative papers are listed here. 
Jackson, J.: Mon. Not. R.A.S. 83, 4, 1922. 
Lundmark, K., and W. J. Luyten: On the Relation between Absolute 

Magnitude and Spectral class as Derived from Observations of Double 

Stars, A.J. 35, 93, 1923. 
Luyten, W. J.: On Some Statistical Properties of Double Stars in Space, 

Proc. Nat. Acad. Set. 16, 252, 1930. 
Gyllenberg, W. : The Binary Stars and the Stream Motions, Med. Lunds. 

Obs. Ser. 1, No. 132, 1932. 
Also a series of papers by E. A. Kreiken in the Monthly Notices. 



CHAPTER XI 

THE ORIGIN OF THE BINARY STARS 

The problem of the origin of the binary stars may be con- 
sidered from two quite different points of view. We may ques- 
tion by what conceivable process or combination of processes 
a single star or, alternately, a primal nebulous mass, can develop 
into a binary system of any kind whatever; or, we may pass in 
review the whole vast series of spectroscopic and visual double 
and multiple systems and ask what theory is competent to 
account not only for the origin of any one binary system but 
also for the great variety in content, form, and dimensions the 
known systems actually exhibit. Obviously, the second point 
of view makes the severer demand upon any theory that may be 
propounded, whether we regard all binaries as objects of a single 
genus, as I am disposed to do, or separate the very close, short- 
period spectroscopic binaries from those of longer period and 
from the visual binaries. 

We may summarize briefly the more important conclusions 
that have been reached in the discussion of the observations of 
the visual double stars in the northern half of the sky and from 
the study of the known orbits of the visual and spectroscopic 
binary systems presented in the three preceding chapters. 

1. A large proportion of the stars are binary systems. On 
the average, at least one in 18 of those as bright as 9.0 magnitude 
is a binary visible in our telescopes and coming within the 
limits set by the "working definition" of a double star; at least 
one in every three or four of those as bright as 5.5 magnitude is a 
binary revealed by the spectrograph. These are minimum 
values. Both visual and spectroscopic binaries within these 
magnitude limits remain to be discovered, our knowledge of 
the fainter stars is still very incomplete, and the angular distance 
limits set by the "working definition" are necessarily arbitrary. 
Unquestionably, many double stars with greater angular dis- 
tances are binary systems. On the evidence before us, we may 
safely say that one-third, and probably two-fifths of the stars 
are binary or multiple systems. 

273 



274 THE BINARY STARS 

2. A considerable percentage of these systems have three or 
more components. It is well within the truth to say that one 
in 20 of the known visual binaries has at least one additional 
member either visible or made known by the spectrograph, and 
many systems are quadruple or even more complex. Many 
of the purely spectroscopic systems are also triple or multiple. 
It is a fact of undoubted significance that, as a rule, triple 
systems, whether visual or spectroscopic, consist of a close binary 
pair and a companion relatively distant. 

3. The masses of the spectroscopic binaries of Class B are 
decidedly greater than those of the spectroscopic binaries of 
later classes, and, among the double-line binaries (i.e., those in 
which the spectra of both components are visible) there is some 
evidence of a progressive decrease in the average mass values 
with advancing spectral class. Evidence of the same kind, 
though less marked, exists for the visual binaries for which 
both orbits and parallax values are available, for the average 
masses of the visual binaries of classes A, F, and G agree very- 
well with those of the spectroscopic binaries of the corresponding 
classes. It is to be noted, however, that double-line spectro- 
scopic binaries and visual binaries with known orbits and 
parallaxes are necessarily selected systems. 

4. The rule, both in visual and in spectroscopic binaries, is 
that the fainter component is the less massive one, but the mass 
ratio is rarely less than one-half and the average, as far as can be 
determined from the available data, is about three-fourths. 

5. Spectroscopic binaries are relatively more numerous among 
stars of classes B and A, and visual binaries among stars of 
classes B, F, and G, than among stars of the other spectral 
classes. The small number of systems of either type among 
stars of Class M is specially striking. 

6. When the primary star of a visual binary is a giant of 
spectral class later than F0, the companion belongs to an earlier 
spectral class; when the primary is a star of the main sequence 
(i.e., a giant of spectral class earlier than F0, or a dwarf) the 
companion belongs to a later spectral class, the difference in 
spectral class, in both cases, increasing with the difference in 
magnitude of the two components. 

7. There is a close correlation between the length of period, 
or size of system, and the degree of ellipticity in the orbit. The 
visual binaries, with periods to be reckoned in years or even in 



THE ORIGIN OF THE BINARY STARS 275 

centuries, have an average eccentricity slightly above 0.5; the 
spectroscopic binaries, with periods to be reckoned, for the most 
part, in days or even in fractions of a day, have an average 
eccentricity of less than 0.2, and in each class the average eccen- 
tricity increases with the average length of period. We have, 
apparently, one unbroken progression or series of orbits from 
systems in which the two components revolve in a fraction of a 
day in circular orbits and practically in surface contact, to 
systems in which the components, separated by one, two, or 
many hundreds of times the distance from the Earth to the 
Sun, revolve in highly elliptic orbits in periods of hundreds and 
even of thousands of years. The scatter of the values of e in 
any given period group is large, but, statistically, we may regard 
the correlation as securely established. 

These facts and relations, as well as others less conclusively 
demonstrated as yet, must all be taken into account when we 
look at the question of the origin of the binary stars from the 
second point of view indicated in the opening paragraph of this 
chapter. We have now to see whether any of the theories so 
far advanced are satisfactory from this point of view. At least 
three theories have been developed that merit consideration. 

The capture theory, apparently first advanced by Dr. G. John- 
ston Stoney, in 1867, is based upon the hypothesis that two 
stars, originally independent, might approach each other under 
such conditions that each would be swerved from its path and 
forced to revolve with the other about a common center of 
gravity. This theory, in its original form, has been completely 
abandoned, but the consequences of the near approach of two 
stars have been the subject of extensive discussion in more 
recent years. Chamberlin and Moulton, for example, have 
argued that, under appropriate initial conditions, such an 
approach might result in a system of planets like our own. This 
is probably the best theory that has been developed to account 
for the origin of the solar system, and if it may be accepted, then 
we may agree that the close approach of two stars, under some- 
what different conditions, might result in a planetary system 
in which the disparity in mass between the central star and its 
largest planet would be far less than it is between the Sun and 
Jupiter. That is the assumption MacMillan* makes in arguing 

* Science 62, 63, 96, 121, 1925. 



276 THE BINARY STARS 

that, in certain cases, a planetary system might ultimately 
develop into a binary star system. 

Assume, for example, that, as the solar system traverses space, 
the Sun and the planets grow in mass from the infall of atoms, 
molecules or larger particles, sometimes very slowly, at others 
(as the system passes through a region filled with nebulous 
matter) more rapidly, and that, on the whole, the Sun's gain by 
this process is offset by its loss of mass through radiation. Then 
the mass of Jupiter, to consider only the largest planet, would 
steadily increase relatively to that of the Sun and the distance 
between the two bodies would decrease. If, in this manner 
Jupiter were to acquire sufficient mass to become a dwarf red 
star while the Sun just held its own in mass, we should have a 
double star system, the other planets being absorbed, ultimately, 
either by Jupiter or by the Sun. MacMillan describes the 
process in more detail, but in a recent personal letter says that 
Jupiter could not become equal to the Sun in mass before it 
merged with it. Indeed it is doubtful if it could exceed one- 
tenth the mass of the Sun without having the two bodies drawn 
together. 

It is hard, therefore, to see a double star future for the solar 
system, for we have found that even in the closest spectroscopic 
binaries the mass ratio rarely falls below one-half. But even 
if we grant that a binary star might develop, in the course 
of eons, from a planetary system in which the original mass 
distribution was more favorable, the process, at best, could 
account for only an occasional spectroscopic binary with quite 
unequal components. Once formed, there would, it is true, be a 
tendency toward equality in mass of the two components, as 
time went on, for the rate of radiation from a star is proportional 
to a power higher than the first,* but we have already noted 
MacMillan's conclusion that in a system so formed the two 
stars are likely to fall together when the mass ratio approaches 
unity. If we assume this danger to be averted through tidal 
interaction and loss of mass by both bodies through radiation, 
then, statistically, the masses of the visual binaries with periods 
measured in decades or in centuries should be smaller than those 
of the spectroscopic binaries, with periods measured in days. 
This, however, we have shown not to be the case (Tables 10 and 

* Jeans, M on. Nat. R.A.S. 85, 209, 1925. 



THE ORIGIN OF THE BINARY STARS 277 

11, Chap. VIII). We must, I fear, decide that the theory does 
not meet the requirements. 

The fission theory which we have next to examine, assumes 
that a star in its primal nebulous stage, or possibly at a later 
one, divides under the stress of its own gravitation, radiation 
pressure, and rotational forces or under the strain of some 
external disrupting force or forces. 

The behavior of a rotating, homogeneous, incompressible, 
fluid mass, in equilibrium and free from external disturbance, 
was investigated by Maclaurin and Jacobi about a century 
ago and later by Poincare\ G. H. Darwin, Liapounoff, Jeans, 
and others. It was found possible, under certain assumptions, 
to follow, mathematically, the transformations of figure as 
the rotating mass contracts under its own gravitation and heat 
radiation from the initial sphere through a succession of spheroids 
and ellipsoids until a pear-shaped figure is reached. It seems 
probable, though it could not be demonstrated mathematically, 
that fission into two masses would follow, the masses revolving 
at first in surface contact and in circular orbits. 

The stars and, in all probability, the antecedent nebulae, 
are neither homogeneous nor incompressible, but it has been 
argued, first, I believe, by See, and later by Darwin and Jeans, 
that a nebula (or even a star in its earliest stage) might none 
the less pass through a similar series of changes and ultimately 
form a stable double star system. Once formed, the forces of 
tidal interaction and of the disturbances ("knocking about") 
produced by the attractions of other stars, were invoked to 
account for the development of the systems with longer periods 
and elliptic orbits. 

Sir James Jeans, in particular, has advocated this theory, 
and readers are referred to his writings cited at the end of the 
chapter for his mathematical development of it. But Jeans 
himself has confirmed Liapounoff's conclusion* that the pear- 
shaped figure is unstable and has pointed out that if a double 
star system results, it, too, will be unstable unless the mass ratio 
of the two components is less than one-third. It has also been 
shown by Moulton, Russell, and Jeans that even were a stable 
double star system to result from fission, the mutual tidal actions 

* Liapounoff, Mem. Imp. Acad. Set. St. Petersburg 17, 1905; 
Jeans, Phil. Trans. Roy. Soc. A 217, 1, 1917. 



278 THE BINARY STARS 

of the two components could never greatly increase either the 
major axis or the eccentricity of the orbit. Quite recently 
William Markowitz* made a statistical investigation of the 
possibility that short-period spectroscopic binaries might develop 
into long period systems and in his work he had the benefit 
of advice from Profs. MacMillan, Moulton, and Otto Struve, 
He found that the necessary increase in the values of P, a, and e 
could not result from contraction, secular decrease of mass, or 
close encounters with other stars nor yet from the combination 
of all three. Jeans had earlier concluded that pairs with periods 
in excess of about 55 days could not result from the process of 
fission, and that a different origin must be sought for them. 

A separation of the binaries at the 55-day period point or at 
any other is negatived, in my judgment, by the fact that there 
is no apparent correlation between period and mass,f and also 
by the fact that there is a definite correlation between eccen- 
tricity and period. If the fission theory is incompetent to 
explain the origin of the longer period binaries, that fact alone 
leads us to abandon it entirely. 

The separate nuclei theory remains to be considered. This 
was first suggested by Laplace more than a hundred years ago. 
In Note VII to his Systeme du Monde, he remarks : 

"Such groups (as the Pleiades) are a necessary result of the 
condensation of nebulae with several nuclei, for it is plain that 
the nebulous matter being constantly attracted by these different 
nuclei must finally form a group of stars like the Pleiades. The 
condensation of nebulae with two nuclei will form stars in very 
close proximity, which will turn one around the other similar to 

* A p. Jour. 76, 69, 1932. 

t In 1924, 1925, Jeans and E. W. Brown both found that, theoretically, 
the semimajor axis (a) and therefore the period (P) would increase if one 
or both components of a binary system were losing mass by radiation. 
Jeans found that the eccentricity (e) would also remain constant, whereas 
Brown's conclusion was that it, too, would increase though at a lower rate. 
In a note in Nature for April 21, 1934, A. E. H. Bleksley states that a 
recent investigation of his own confirms Jeans's results. He concludes 
that the "semiaxis major is inversely proportional to the mass of the system" 
throughout the life of a binary. He adds that a statistical study of all 
available material shows that this relation appears to hold for the visual 
binaries of known orbit but not for the short-period spectroscopic binaries, 
and he suggests that there is a difference in origin between the two groups. 
No details are given. 



THE ORIGIN OF THE BINARY STARS 279 

those double stars whose relative motions have already been 
determined."* 

The modern writer who has adopted this theory most explicitly 
is the very man who first formulated the fission theory, Dr. See. 
His discussion of the binary stars in the second volume of his 
Researches on the Evolution of the Stellar System is devoted more 
particularly to the development of the systems after their initial 
formation but on page 232 we find the statement, "It is evident 
. that the resulting mass-ratio in a system depends on the 
supply of nebulosity and the original nuclei already begun and 
slowly developing in the nebula while it was still of vast extent 
and great tenuity," and, on page 584, the even more definite 
statement, "When a double star had been formed in the usual 
manner by the growth of separate centers in a widely diffused 

nebula. ..." 

The separate nuclei theory apparently affords sufficient 
latitude for the explanation of any binary system except, perhaps, 
the very close, short-period spectroscopic binaries. To account 
for these, the effect of a resisting medium has been invoked, 
Markowitz, for example, finding that, unless the two components 
are radiating mass more than 2.5 as fast as they are gathering 
it in, the values of P, a, and e of the system are decreasing. 
But even if we grant a tendency toward such a decrease, it is 
hardly conceivable that it would account for the transformation 
of, say, hundred-year-period systems into systems with periods 
of a single day, or of ten days, especially in view of the fact that 
no correlation appears to exist between mass and period. 

The fundamental objection to the separate nuclei theory is 
that we really do not explain anything; to use Moulton's words, 
"we only push by an assumption the problem of explaining the 
binary systems a little farther back into the unknown." 

Russell's specific objection, based upon his study of the 
triple and quadruple systems, is also of great force. Why 
should these systems almost invariably consist of a compara- 
tively close binary pair attended by a third star or by another 
close binary pair at a distance relatively great? As Russell 
says, "Not only is there no apparent reason for it, but if we 
try to retrace in imagination the history of such a system, 
through stages of greater and greater diffusion as we penetrate 

*See Essays in Astronomy, p. 501 (edited by E. S. Holden, New York, 
D. Appleton A'Company, 1900). 



280 THE BINARY STARS 

farther into the past (keeping in mind that the moment of 
momentum of the whole system must remain constant), it is 
hard to form any idea of the history of the nuclei which will 
finally form a close and rapidly revolving pair, attended by a 
distant companion." 

The general conclusion of our discussion is that, although the 
observational data that have been accumulated clearly indicate 
the common origin of the binary stars, no theory of that origin 
and of the subsequent development of the observed systems 
that has so far been formulated can be regarded as satisfactory. 

It is the duty of the observers to supply us with the data 
required for the formulation and test of such a theory. Observa- 
tions of the various classes of binary systems, visual, spectro- 
scopic and eclipsing, must be continued indefinitely, by the 
methods described in earlier chapters or by improved methods, 
to provide the data needed for orbit computation. More orbits 
of all these classes of systems are, of course, demanded. We 
must have a number sufficiently great to insure confidence that 
conclusions based upon our discussions are not affected by 
selection in the data. 

But that is not all. We are quite as much in need of accurate 
values of the parallaxes of the systems and of accurate deter- 
minations of the magnitudes and spectral classes of the fainter 
as well as of the brighter components. A knowledge of the 
masses, the mass ratios, the absolute magnitudes, and the 
spectral characteristics of the components in these systems is 
quite as important as a knowledge of the orbital elements for the 
solution of the problem of the origin and evolution of the binary 
stars. 

References 

The following list contains only a few of the many papers that have been 

read in preparing this chapter. References to others will be found in them, 

and also in some of the papers cited at the end of Chap. VIII. 

Darwin, G. H. : The Genesis of Double Stars, Darwin and Modern Science, 
pp. 543-564. Cambridge University Press, 1910. 

: Presidential Address, British Association for the Advancement 

of Science, Report B. A. A. S. 1905, p. 3. 

Jeans, J. H. : The Motion of Tidally Distorted Masses with Special Refer- 
ence to Theories of Cosmogony. Mem. R.A.S. 62, part 1, 1917. 

: On the Density of Algol Variables, Ap. Jour. 22, 93, 1905. 

: The Evolution of Binary Systems, Mon. Not. R.A.S. 79, 100, 1918. 

: The Origin of the Binary Stars, Scientia 31, 11, 1922. 



THE ORIGIN OF THE BINARY STARS 281 

: Astronomy and Cosmogony, pp. 198-307, Cambridge University 

Press, 1928. . 

Moulton, F. R.: On Certain Relations among the Possible Changes in the 
Motions of Mutually Attracting Spheres when Disturbed by Tidal 
Interactions; and, Notes on the Possibility of fission of a Contracting 
Rotating Fluid Mass, Publication 107, Carnegie Institution of Washing- 
ton, pp. 77-160. 

Introduction to Astronomy, revised edition, pp. 543-548 



Russell, H. N.: On the Origin of Binary Stars, Ap. Jour. 31, 185, 1910. 
See, T. J. J.: Die Entwickelung der Doppelstern-Systeme, Inaugural 

Dissertation, 1892. 
: Researches on the Physical Constitution of the Heavenly Bodies. 

A.N. 169, 321, 1905. 
-: Researches on the Evolution of the Stellar Systems, 2, Chap. 20, 1910. 



MacMillan, W. D.: The Problem of Two Bodies with Diminishing Mass, 

Mon. Not. R.A.S. 86, 904, 1925. 
Shajn, G.: On the Mass-Ratio in Double Stars, Mon. Not. R.A.S. 86, 245, 

1929; A.N. 287, 57, 1929. 
Markowitz, W. : The Evolution of Binary Stars, Ap. Jour. 76, 69, 1932. 
: Some Statistical and Dynamical Aspects of the Fission Theory, 

Ap. Jour. 78, 161, 1933. 



APPENDIX 

The two tables printed on the following pages list the visual 
and spectroscopic binary star orbits that have been used in the 
preparation of the present volume. In compiling them, all 
orbits available to me by September, 1933, have been examined. 
The visual orbits listed in Table I have been taken from my 
card catalogue of orbits, the orbits of the spectroscopic binaries, 
from the card catalogue of variable radial velocities which 
Dr. J. H. Moore keeps up to date.* I am deeply indebted 
to Mrs. Moore for her kindness in copying off the necessary 
data. Both sets of orbits have been checked by examination 
of the original publications. 

The abbreviations used for publications cited in the footnotes 
and at the end of the chapters in the volume will, I think, be 
readily understood. In the following tables, however, it has been 
necessary to use the shortest possible form, and it may be well to 
state that A.N., A.J., Ap. Jour., M.N., P.A., B.A.N., P.A.S.P., 
and R.A.S.C., stand, respectively, for AstronomischeNachrichten, 
Astronomical Journal, Astrophysical Journal, Monthly Notices of 
the Royal Astronomical Society, Popular Astronomy, Bulletin 
of the Astronomical Institutes of the Netherlands, Publications 
of the Astronomical Society of the Pacific, and Journal of the 
Royal Astronomical Society of Canada. "Observatory," has been 
abbreviated to 0., D.A.O., D.O. and A.O. standing, respectively, 
for the Dominion Astrophysical, the Dominion and the Allegheny 
Observatories. C.A. represents the Cape (of Good Hope) 
Annals. The other abbreviations need no explanation. 



* [Dr. Moore died on March 15, 1949. See p. 40.— J.T.K.] 

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THE BINARY STARS 




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NAME INDEX 



Adams, W. S., 228, 239, 242, 248 

Airy, G., 17 

Alter, D., 121, 244 

Anderson, J. A., 68, 250 

Andre, C, 123 

Argelander, F. W. A., 264 

Astrand, J. J., 80 

Auwers, A., 118, 237, 238, 243 



B 



Baize, P., 40, 244 

Baker, R. H., 182 

Banachiewicz, T., 202 

Barnard, E. E., 23, 47, 60, 240, 241 

Barr, J. M., 213 

Barton, S. G., 67 

Bell, L., 270 

Belopolsky, A., 134, 248, 254 

Bergstrand, O., 244 

Berman, L., 247, 248 

Bernewitz, E., 97 

Bessel, F. W., 13, 19, 118, 119n., 
225, 243 

Bishop, G., 17 

Bleksley, A. E. H., 278 

Bode, J. E., 3, 4 

Bohlin, K., 215 

Bond, G. P., 1, 64, 237 

Boothroyd, S. L., 22 

Bos, W. H. van den, 37, 39, 60n., 
71, 78n., 79, 90, 112, 119, 204, 
209n., 240, 246, 247, 260 

Bosler, J., 62 

Boss, L., 226, 227, 239 

Bradley, J., 1, 2, 14, 243 

Brashear, J. A., 37 

Brill, A., 234 

Brisbane, T. M., 35 



Brown, E. W., 278n. 
Burnham, S. W., 10, 17, 19-22, 24, 
25, 32-35, 53, 59, 238, 240, 241 



Campbell, W. W., 29, 46n., 135, 

145, 150, 206, 210, 219, 249 
Cannon, Annie J., 130, 267 
Chamberlin, T. C., 275 
Chandler, S. C, 182, 254 
Chang, Y. C, 216 
Clark, A., 17, 20, 23 
Clark, A. G., 23, 118, 225, 237 
Cogshall, W. A., 22 
Comstock, G. C, 72, 111, 117, 120, 

124, 228, 231 
Cornu, M. A., 142, 149 
Crawford, R. T., 114, 115 
Crossley, E., 32 
Curtis, H. D., 29, 159, 162, 165, 180, 

248 
Curtiss, R. H., 146, 150, 180, 254. 

255 



D 



Danjon, M. A., 256n. 

Darwin, G. H., 277, 280 

Dawes, W. R., 17, 53, 57, 63, 72 

Dawson, B. H., 37, 124, 244 

De Sitter, W., 124 

Dembowski, Ercole, 18, 19, 53, 72 

Ditscheiner, L., 145 

Doberck, W., 124, 203, 205, 206, 215, 

235 
Doig, P., 270n. 
Donner, H. F., 38 
Doolittle, Eric, 33, 34, 241 
Doppler, Ch., 125 



303 



304 



THE BINARY STARS 



Dugan, R. S., 182, 198n. 
Dunlop, J., 35 

E 

Eberhard, G., 66, 150 

Eddington, A. S., 229, 233, 251, 252 

Elkin, W. L., 47, 236 

Encke, J. F., 71 

Englemann, R., 19 

Espin, T. E. H., 23, 34 

Everett, Alice, 215 



Fetlaar, J., 202 

Feuille, L., 2 

Finsen, W. S., 37, 102, 104, 105, 216, 

235, 236, 240 
Fizeau, H. L., 125 
Flammarion, C, 32 
Fletcher, A., 73n., 244 
Fontenay, J. de, 1 
Fox, Philip, 33, 64, 240 
Fraunhofer, J., 11 
Frost, E. B., 145, 150 
Furner, H. F., 228 



Harting, C. A. J., 183 

Hartmann, J. F., 142, 147, 150, 251 

Hartwig, E., 202 

Heger, M. (Mrs. C. D. Shane), 252 

Hellerich, J., 254, 256 

Henroteau, F. C, 40, 114, 124, 175, 

180 
Henry, P., 64 
Herschel, John, 9, 10, 15, 20, 25, 

32-36, 41, 71, 72, 76, 235, 243, 

246 
Herschel, William, 3-13, 15, 20, 25, 

32, 33, 41, 70, 232, 246 
Hertzsprung, E., 64-66, 228, 231, 

233, 246, 247, 271 
Holden, E. S., 23, 33, 279». 
Hooke, R., 1 
Hough, G. W., 22, 32 
Howard, C. P., 71, 123 
Howe, H. A., 23 
Huggins, William, 134 
Humasson, M. L., 252 
Humboldt, A. von, 237 
Hussey, W. J., 17, 24, 25, 33, 37, 38, 

57, 232, 245, 258 
Huygens, C, 1 



G 



Galileo, G., 5 

Gaposchkin, S., 201, 202 

Gerasimovic, B. P., 252 

Gill, David, 42, 236 

Glasenapp, S., 71, 80, 89, 95, 97, 101 

Gledhill, J., 19, 32 

Goodericke, J., 27, 254 

Gould, B. A., 64 

Gyllenberg, W., 272 



Innes, R. T. A., 36-39, 54, 71, 79, 
80, 90, 91, 94, 107, 111, 237, 240 



Jacobi, K. G. J., 277 
Jackson, J., 228, 231, 233, 272 
Jeans, J., 277, 280 
Jessup, M. K., 37 
Jonckheere, R., 23, 34 
Joy, A. H., 218n., 228, 248 



Hagen, J. G., 202 
Hall, Asaph, 31, 32, 123 
Halley, E., 2 
Halm, J. K. E., 180 
Hargrave, L., 36 
Hargreaves, F. J., 41n., 67 
Harper, W. E., 177n. 



K 

Keeler, J. E., 150 

Kepler, J., 93, 106, 113, 204 

Kerrich, J. E., 124 

King, E. S., 180 

King, W. F., 157n., 169, 180 

Kirchhoff, G. R., 129 



NAME INDEX 



305 



Klinkerfues, E. F. W., 71, 123 

Knott, G., 19 

Kowalsky, M., 71, 80, 82, 89, 95, 97 

Krat, W., 202 

Kreiken, E. A., 215, 234, 272 

Krueger, F., 241 

Kuiper, G. P., 205n., 238, 242, 271 

Kustner, F., 23, 33 



Lacaille, N. L. de, 235 

Lambert, J. H., 2 

Lamont, R. P., 37 

Laplace, P. S., 278 

Lau, H. A., 270 

Laves, K., 114, 175, 179 

Leavenworth, F. P., 97 

Lehmann-Filhes, R., 30, 158-165, 

179, 225 
Leonard, F. C, 270 
Leuschner, A. O., 70, 123 
Lewis, T., 14, 15, 19, 33, 57, 58, 61, 

121, 203, 215 
Liapounoff, A. M., 277 
Lohse, O., 235 

Ludendorff, H., 206, 214, 233 
Lundmark, K., 234, 272 
Luplau-Janssen, C, 204 
Luyten, W. J., 248, 272 

M 

McClean, F., 36 
McDiarmid, R. J., 196 
McLaughlin, D. B., 255, 256 
Maclaurin, C., 277 
MacMillan, W. D., 275-278, 281 
Maderni, A., 180 
Madler, J. H., 19, 71 
Maggini, M., 68, 69, 97 
Markowitz, W., 278, 279, 281 
Maury, Antonia C, 28, 130 
Maxwell, A. D., 245ro. 
Mayer, C., 3, 4 
Mayer, T., 232 
Merrill, P. W., 68, 250, 252 
Meyermann, B., 124 
Michell, J., 2, 8 



Michelson, A. A., 68, 250 

Milburn, W., 23 

Montanari, G., 253 

Moore, Charlotte E., 230, 231 

Moore, J. H., 29, 125, 155n., 180, 239 

Moulton, F. R., 114, 275, 278, 281 

Miiller, G., 202 

N 

Nassau, J. J., 124 
Newall, H. F., 150, 249 
Newton, Isaac, 70 
Nijland, A. N., 179, 182 
Norlund, N. E., 119, 246 



O 



Olivier, C. P., 67 
Oort, J. H., 271 
Orloff, A., 180 



Paddock, G. F., 177n., 179, 180 

Parkhurst, J. A., 182 

Parvulesco, C, 124 

Pearce, J. A., 252, 253 

Peters, C. A. F., 118, 237 

Peters, C. F. W., 244 

Piazzi, G., 243 

Picart, L., 180 

Pickering, E. C, 1, 26, 64, 130, 182, 

183, 201, 254, 258». 
Pitman, J. H., 234 
Plaskett, J. S., 150, 252 
Plummer, H. t 179, 180 
Pogo, A., 180, 214 
Poincare, H.,- 277 
Pollock, J. A., 36 
Poor, J. M., 216 
Pound, J., 1, 2, 245 
Prager, R., 202 
Przybyllok, E., 246 
Ptolemy, 2 



R 



Rambaut, A. A., 30, 123, 164, 179 
Redman, R. O., 231 



306 



THE BINARY STARS 



Reese, H. M., 250 

Riccioli, J. B., 1 

Richaud, J., 1 

Roberts, A. W., 182, 183, 201 

Rossiter, R. A., 37, 255 

Rowland, H. A., 141, 146 

Russell, H. C, 36 

Russell, H. N., 71, 80, 86, 117, 118, 
124, 164, 179, 183, 184, 188, 
192, 197, 201, 208, 226n., 228- 
231, 277, 279, 281 



S 



Safford, T. H., 237 

Salet, P., 62 

Sanford, R. S., 248 

Savary, F., 71 

Schaeberle, J. M., 118 

Scharbe, 202 

Scheiner, J., 67, 134 

Schiaparelli, G. V., 19, 31, 59 

Schlesinger, F., 63, 121, 155, 175n., 

176, 180, 206, 214, 223, 236, 

244, 245n., 254, 255 
Schnauder, M., 216 
Schorr, R. H., 123 
Schuster, A., 56 
Schwarzschild, K., 68, 71, 123, 156, 

165, 168, 179 
Seares, F. H., 182, 242 
Secchi, A., 19, 130 
See, T. J. J., 22, 71, 108, 205, 215, 

235, 277, 279, 281 
Seeliger, H. von, 71, 119, 123, 260- 

262 
Sellors, R. P., 36 

Shajn, G., 215, 233, 234, 270»., 281 
Shapley, H., 183, 197, 201, 202 
Sitterly, B. W., 183, 202 
Slipher, V. M., 252 
Smythe, W. H., 18 
South, J., 9, 10, 243w. 
Stebbins, Joel, 182, 231, 255 
Stein, J., 67, 202 
Stone, O., 23 
Stoney, G. J., 275 



Struve, F. G. W., 11-20, 25, 33, 36, 

72, 246, 249, 270 
Struve, H:, 62 
Struve, O., 17, 19, 24, 25, 33, 61, 

114, 245 
Struve, Otto, 204, 205, 214, 252 



Thiele, T. N., 71, 79, 89-92, 94, 101, 

107, 111, 123 
Tisserand, F. F., 183 
Turner, H. H., 215 



Vahlen, T., 124 

Van Biesbroeck, G., 33, 97, 107 

Van Gent, H., 248 

Villarceau, Yvon, 71 

Vogel, H. C., 27, 28, 134, 150, 254 

Vogt, H., 202 

Volet, C., 80n., 124, 238, 240, 241 

Voute, J., 62 

Vyssotsky, A., 238, 240 

W 

Wallenquist, A., 271 
Walter, K., 202 
Webb, T. W., 21, 22 
Wendell, O. C., 182, 192, 238 
Wilkins, P. O., 124 
Wilsing, J., 164, 179 
Wilson, J. M., 19, 32 
Wright, W. H., 119, 136, 138, 150, 
225, 236, 247, 256 



Young, R. K., 154n., 155n., 173n. 

Z 

Zagar, F., 240 

Zurhellen, W., 156, 165, 179 

Zwiers, H. J., 71, 80, 86, 95, 100, 101 



SUBJECT INDEX 



Algol, 253 

Angstrom unit defined, 129 

B 

Binary stars, definition of, ix 
density of, 231 
distribution of, by spectral class, 

267-269 
invisible companions in systems 

of, 118, 225 
masses of, 216 
multiple systems of, 224, 232, 245, 

274 
number of known orbits of, 203- 

205 
parallaxes of, from orbits and 

radial velocities, 224 
relations between period, and 
eccentricity in orbits of, 205, 
213 
and spectral class in orbits of, 
210 
systems of special interest, 235 
tables, of the known orbits of, 
283, 288 
of masses of, 220 
theories of the origin of, capture 
theory, 275 
fission theory, 277 
general statement of problem 

of, 273 
separate nuclei theory, 278 
summary of, 280 
(See also Double stars; Eclipsing 
binary stars; Spectroscopic 
binary stars; Visual binary 
stars) 



Cancri, Tau, 232 

CapeUa, 249 

Castor, 248 

Centauri, Alpha, 235 

Cepheid variable stars, 30 

Color contrast in double stars, 16, 

270 
61 Gygni, 243 



D 



Doppler-Fizeau principle, 126 
Double stars, Burnham's discoveries 
and measures, 20, 25 
color contrast in, 16, 270 
conventions for measures of, 15, 41 
correction for proper motion in, 73 
diaphragms used in measuring, 59 
distribution of, by angular dis- 
tances, and magnitudes, etc., 
261 
in right ascension and declina- 
tion, 260 
early discoveries of, 1 
early orbit methods for, 71, 72 
early speculations on character of, 

2 
errors of measure of, 60 
eyepieces for measuring, 59 
first collection of, 3 
first photograph of, 1, 64 
general catalogues of, 32, 39 
Herschel's discoveries and theo- 
ries, 4 
Herschel (J) and South's work on, 

9 
interferometer measures of, 67 
Lick Observatory survey for dis- 
covery of, 24, 257 



307 



308 



THE BINARY STARS 



Double stars, magnitude estimates 
of components of, 54 
methods of measuring, 49 
observing program for, 55 
percentage of, among stars near 
the Sun, 267 
by magnitude classes, 265 
photographic measures of, 64 
southern hemisphere work on, 35 
Struve (F. G. W.) and the 
Mensurae Micrometricae, 12 
Struve (O.) and the Pulkowa 

Catalogue, 17 
working definition of, 257 

(See also Binary stars; Visual 
binary stars) 
Driving clock, the first, 11, 12 
Dynamical parallaxes, 227 



E 



Eclipsing binary stars, density of, 
198 
early history of, 181 
example of orbit of, 192 
reflection effect in, 198 
Russell's method of computing the 

orbits of, 183 
table of orbits of, 200 
Equulei, Delta, 244 
Eyepieces, methods of finding the 
magnifying power of, 59 



G 



Gravitation, universality of the law 
of, 70 

H 

Hercvlis, Zeta, 120 



Interferometer measures of double 
stars, 67 

K 

Kirchhoff's law, 129 
Krueger 60, 241 



lick Observatory double star survey, 
24, 257 



M 



Masses, 216 

table of, 220 
Mass function, 217-223 
Micrometer, the comparison image, 
41 
the filar, 41 
Micrometer screw, value of revolu- 
tion of, 44 
Multiple systems, 120, 224, 232, 245, 
274 

O 

70 Ophiuchi, 120 
Orbit of A 88, 95 

of 24 Aqr., 101 

of 42 C. Br., 113 

of W Del., 192 

of € Equ., 117 

of k Vel., 162 



Parallel, method of determining, 43 
Photographic methods of measuring 
double stars, 64 



R 



Radial velocity of a star, chapter on 
the, by J. H. Moore, 125 
early determinations of the, 134 
tests for correctness of measures 
of, 149 
(See also Spectrograms; Spectro- 
graphs; Spectroscopes) 
Rectilinear motion, 121 
Resolving power, of a spectroscope, 
128 
of a telescope, 56 



S 



Sirius, 237 



SUBJECT INDEX 



309 



Spectrograms, Cornu-Hartmann for- 
mulae for, 142 
dispersion curves for reducing, 142 
example of the measurement and 

reduction of, 141 
spectro-comparator method of re- 
duction of, 147 
velocity-standard method of re- 
duction of, 146 
Spectrograph, description of the 
Mills, 135 
loss of light in a stellar, 139 
Spectroscope, essential parts of, 127 

resolving power of, 128 
Spectroscopic binary stars, cata- 
logues of, 29 
differential correction of the ele- 
ments of, 175 
discoveries of, 1, 26, 39 
distribution of, by spectral class, 
269 
co in orbits of, 213 
equation for the radial velocity of, 

152 
example of orbit determination of, 

162 
first solution of the orbit problem, 

30 
H and K lines in, 251 
masses of the, 217 
methods of orbit computation, 
conditions of the problem, 
151 
fundamental formulae, 153, 157 
King's method, 169 
Lehmann-Filhes's method, 158 
Russell's short method, 174 
Schwarzschild's method, 165 
Zurhellen's methods, 166 
orbits of, with small eccentricity, 

178 
secondary oscillations in orbits of, 
177 



Spectroscopic binary stars, table 
of orbits of, 288 
velocity curve for, 154, 157 
(See also Binary stars; Eclipsing 
binary stars) 
Stellar radial velocities, determina- 
tion of, 134 
Stellar spectra, classification of, 130 



U 



Ursae Majoris, Xi, 119, 246-248 



Variable proper motion, 118, 237 
Visual binary stars, apparent ellipse, 
from elements of, 108 
from observations of, 72 
definition of orbit elements of, 77 
differential corrections to elements 

of, 109 
distribution of longitudes of peri-, 

helion in, 213 
dynamical parallaxes of, 227 
formulae, for ephemerides for, 79, 
80 
for orbits when i = 90°, 113 
mass ratios in, 226 
methods for computing orbits of, 
Glasenapp-Kowalsky method, 
80,95 
special methods, 113 
Thiele-Innes constants and 

methods, 89, 101 
Zwiers's method, 86, 100 
orientation of the orbit planes of, 

215 
statistical data relating to the, 257 
table of orbits of the, 284 
(See also Double stars) 



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which provides excellent innoculation. "Should be read by everyone, scientist or non- 
scientist alike," R. T. Birge, Prof. Emeritus of Physics, Univ. of Calif; Former Pres., 
Amer. Physical Soc. x + 365pp. 5% x 8. T394 Paperbound SI .50 

ON MATHEMATICS AND MATHEMATICIANS, R. E. Moritz. A 10 year labor of love by discerning, 
discriminating Prof. Moritz, this collection conveys the full sense of mathematics and 
personalities of great mathematicians. Anecdotes, aphorisms, reminiscences, philosophies, 
definitions, speculations, biographical insights, etc. by great mathematicians, writers: Des- 
cartes, Mill, Locke, Kant, Coleridge, Whitehead, etc. Glimpses into lives of great mathema- 
ticians, from Archimedes to Euler, Gauss, Weierstrass. To mathematicians, a superb 
browsing-book. To laymen, exciting revelation of fullness of mathematics. Extensive cross 
index. 410pp. 5% x 8. T489 Paperbound SI. 95 

GUIDE TO THE LITERATURE OF MATHEMATICS AND PHYSICS, N. G. Parke III. Over 5000 
entries under approximately 120 major subject headings, of selected most important books, 
monographs, periodicals, articles in English, plus important works in German, French, 
Italian, Spanish, Russian, (many recently available works). Covers every branch of physics, 
math, related engineering. Includes author, title, edition, publisher, place, date, number 
of volumes, number of pages. 40 page introduction on basic problems of research, study 
provides useful information on organization, use of libraries, psychology of learning, etc. 
Will save you hours of time. 2nd revised edition. Indices of authors, subjects. 464pp. 
5% x 8. S447 Paperbound $2.49 

THE STRANGE STORY OF THE QUANTUM, An Account for the General Reader of the Growth 
of Ideas Underlying Our Present Atomic Knowledge, B. Hoffmann. Presents lucidly, expertly, 
with barest amount of mathematics, problems and theories which led to modern quantum 
physics. Begins with late 1800's when discrepancies were noticed; with illuminating anal- 
ogies, examples, goes through concepts of Planck, Einstein, Pauli, Schroedinger, Dirac, 
Sommerfield, Feynman, etc. New postscript through 1958. "Of the books attempting an 
account of the history and contents of modern atomic physics which have come to my 
attention, this is the best," H. Margenau, Yale U., in Amer. J. of Physics. 2nd edition. 32 
tables, illustrations. 275pp. 5% x 8. T518 Paperbound $1.45 



DOVER SCIENCE BOOKS 

HISTORY OF SCIENCE 

AND PHILOSOPHY OF SCIENCE 

THE VALUE OF SCIENCE, Henri Poincar*. Many of most mature ideas of "last scientific 
universalist" for both beginning, advanced workers. Nature of scientific truth, whether 
order is innate in universe or imposed by man, logical thought vs. intuition (relating to 
Weierstrass, Lie, Riemann, etc), time and space (relativity, psychological time, simultaneity), 
Herz's concept of force, values within disciplines of Maxwell, Carnot, Mayer, Newton, 
Lorentz, etc. iii + 147pp. 5% x 8. S469 Paperbound $1.35 

PHILOSOPHY AND THE PHYSICISTS, L. S. Stebbing. Philosophical aspects of modern science 
examined in terms of lively critical attack on ideas of Jeans, Eddington. Tasks of science, 
causality, determinism, probability, relation of world physics to that of everyday experience, 
philosophical significance of Planck-Bohr concept of discontinuous energy levels, inferences 
to be drawn from Uncertainty Principle, implications of "becoming" involved in 2nd law 
of thermodynamics, other problems posed by discarding of Laplacean determinism. 285pp. 
5% x 8. T480 Paperbound SI .65 

THE PRINCIPLES OF SCIENCE, A TREATISE ON LOGIC AND THE SCIENTIFIC METHOD, W. S. 
Jevons. Milestone in development of symbolic logic remains stimulating contribution to in- 
vestigation of inferential validity in sciences. Treats inductive, deductive logic, theory of 
number, probability, limits of scientific method; significantly advances Boole's logic, con- 
tains detailed introduction to nature and methods of probability in physics, astronomy, 
everyday affairs, etc. In introduction, Ernest Nagel of Columbia U. says, "[Jevons] continues 
to be of interest as an attempt to articulate the logic of scientific inquiry." liii + 786pp. 
5% x 8. S446 Paperbound $2.98 

A HISTORY OF ASTRONOMY FROM THALES TO KEPLER, J. L. E. Dreyer. Only work in English 
to give complete history of cosmological views from prehistoric times to Kepler. Partial 
contents: Near Eastern astronomical systems, Early Greeks, Homocentric spheres of 
Euxodus, Epicycles, Ptolemaic system, Medieval cosmology, Copernicus, Kepler, much more. 
"Especially useful to teachers and students of the history of science . . . unsurpassed in 
its field," Isis. Formerly "A History of Planetary Systems from Thales to Kepler." Revised 
foreword by W. H. Stahl. xvii + 430pp. 5% x 8. S79 Paperbound $1.98 

A CONCISE HISTORY OF MATHEMATICS, D. Struik. Lucid study of development of ideas, 
techniques, from Ancient Near East, Greece, Islamic science, Middle Ages, Renaissance, 
modern times. Important mathematicians described in detail. Treatment not anecdotal, but 
analytical development of ideas. Non-technical — no math training needed. "Rich in con- 
tent, thoughtful in interpretations," U.S. Quarterly Booklist. 60 illustrations including 
Greek, Egyptian manuscripts, portraits of 31 mathematicians. 2nd edition, xix + 299pp. 
5% x 8. S255 Paperbound $1.75 

THE PHILOSOPHICAL WRITINGS OF PEIRCE, edited by Justus Buchler. A carefully balanced 
expositon of Peirce's complete system, written by Peirce himself. It covers such matters 
as scientific method, pure chance vs. law, symbolic logic, theory of signs, pragmatism, 
experiment, and other topics. "Excellent selection . . . gives more than adequate evidence 
of the range and greatness," Personalis! Formerly entitled "The Philosophy of Peirce." 
xvi + 368pp. T217 Paperbound $1.95 

SCIENCE AND METHOD, Henri Poincart. Procedure of scientific discovery, methodology, ex- 
periment, idea-germination — processes by which discoveries come into being. Most signifi- 
cant and interesting aspects of development, application of ideas. Chapters cover selection 
of facts, chance, mathematical reasoning, mathematics and logic; Whitehead, Russell, 
Cantor, the new mechanics, etc. 288pp. 5% x 8. S222 Paperbound $1.35 

SCIENCE AND HYPOTHESIS, Henri Poincart. Creative psychology in science. How such con- 
cepts as number, magnitude, space, force, classical mechanics develooed, how modern 
scientist uses them in his thought. Hypothesis in physics, theories of modern physics. 
Introduction by Sir James Larmor. "Few mathematicians have had the breadth of vision 
of Poincar6, and none is his superior in the gift of clear exposition," E. T. Bell. 272pp. 
5% x 8. S221 Paperbound $1.35 

ESSAYS IN . EXPERIMENTAL LOGIC, John Dewey. Stimulating series of essays by one of most 
influential minds in American philosophy presents some of his most mature thoughts on 
wide range of subjects. Partial contents: Relationship between inquiry and experience; 
dependence of knowledge upon thought; character logic; judgments of practice, data, and 
meanings; stimuli of thought, etc. viii + 444pp. 5% x 8. T73 Paperbound $1.95 

WHAT IS SCIENCE, Norman Campbell. Excellent introduction explains scientific method, role 
of mathematics, types of scientific laws. Contents: 2 aspects of science, science and 
nature, laws of chance, discovery of laws, explanation of laws, measurement and numerical 
laws, applications of science. 192pp. 5% x 8. S43 Paperbound $1.25 



CATALOGUE OF 

FROM EUCLID TO EDDINGTON: A STUDY OF THE CONCEPTIONS OF THE EXTERNAL WORLD, Sir 
Edmund Whittaker. Foremost British scientist traces development of theories of natural phi- 
losophy from western rediscovery of Euclid to Eddington, Einstein, Dirac, etc. 5 major 
divisions: Space, Time and Movement; Concepts of Classical Physics; Concepts of Quantum 
Mechanics; Eddington Universe. Contrasts inadequacy of classical physics to understand 
physical world with present day attempts of relativity, non-Euclidean geometry, space 
curvature, etc. 212pp. 5% x 8. T491 Paperbound $1.35 

THE ANALYSIS OF MATTER, Bertrand Russell. How do our senses accord with the new 
physics? This volume covers such topics as logical analysis of physics, prerelativity 
physics, causality, scientific inference, physics and perception, special and general rela- 
tivity, Weyl's theory, tensors, invariants and their physical interpretation, periodicity and 
qualitative series. "The most thorough treatment of the subject that has yet been pub- 
lished," The Nation. Introduction by L. E. Denonn. 422pp. 5% x 8. T231 Paperbound $1.95 

LANGUAGE, TRUTH, AND LOGIC, A. Ayer. A clear introduction to the Vienna and Cambridge 
schools of Logical Positivism. Specific tests to evaluate validity of ideas, etc. Contents: 
function of philosophy, elimination of metaphysics, nature of analysis, a priori, truth and 
probability, etc. 10th printing. "I should like to have written it myself," Bertrand Russell. 
160pp. 5% x 8. T10 Paperbound $1.25 

THE PSYCHOLOGY OF INVENTION IN THE MATHEMATICAL FIELD, J. Hadamard. Where do ideas 
come from? What role does the unconscious play? Are ideas best developed by mathematical 
reasoning, word reasoning, visualization? What are the methods used by Einstein, Poincare, 
Galton, Riemann? How can these techniques be applied by others? One of the world's 
leading mathematicians discusses these and other questions, xiii + 145pp. 5% x 8. 

T107 Paperbound $1.25 

GUIDE TO PHILOSOPHY, C. E. M. Joad. By one of the ablest expositors of all time, this is 
not simply a history or a typological survey, but an examination of central problems in 
terms of answers afforded by the greatest thinkers: Plato, Aristotle, Scholastics, Leibniz, 
Kant, Whitehead, Russell, and many others. Especially valuable to persons in the physical 
sciences; over 100 pages devoted to Jeans, Eddington, and others, the philosophy of 
modern physics, scientific materialism, pragmatism, etc. Classified bibliography. 592pp. 
5% x 8. T50 Paperbound $2.00 

SUBSTANCE AND FUNCTION, and EINSTEIN'S THEORY OF RELATIVITY, Ernst Cassirer. Two 

books bound as one. Cassirer establishes a philosophy of the exact sciences that takes into 
consideration new developments in mathematics, shows historical connections. Partial 
contents: Aristotelian logic, Mill's analysis, Helmholtz and Kronecker, Russell and cardinal 
numbers, Euclidean vs. non-Euclidean geometry, Einstein's relativity. Bibliography. Index, 
xxi + 464pp. 5% x 8. T50 Paperbound $2.00 

FOUNDATIONS OF GEOMETRY, Bertrand Russell. Nobel laureate analyzes basic problems in 
the overlap area between mathematics and philosophy: the nature of geometrical knowledge, 
the nature of geometry, and the applications of geometry to space. Covers history of non- 
Euclidean geometry, philosophic interpretations of geometry, especially Kant, projective 
and metrical geometry. Most interesting as the solution offered in 1897 by a great mind 
to a problem still current. New introduction by Prof. Morris Kline, N.Y. University. "Ad- 
mirably clear, precise, and elegantly reasoned analysis," International Math. News, xii + 
201pp. 5% x 8. S233 Paperbound $1.60 

THE NATURE OF PHYSICAL THEORY, P. W. Bridgman. How modern physics looks to a highly 
unorthodox physicist — a Nobel laureate. Pointing out many absurdities of science, demon- 
strating inadequacies of various physical theories, weighs and analyzes contributions of 
Einstein, Bohr, Heisenberg, many others. A non-technical consideration of correlation of 
science and reality, xi -I- 138pp. 5% x 8. S33 Paperbound $1.25 

EXPERIMENT AND THEORY IN PHYSICS, Max Born. A Nobel laureate examines the nature 
and value of the counterclaims of experiment and theory in physics. Synthetic versus 
analytical scientific advances are analyzed in works of Einstein, Bohr, Heisenberg, Planck, 
Eddington, Milne, others, by a fellow scientist. 44pp. 5% x 8. S308 Paperbound 60$ 

A SHORT HISTORY OF ANATOMY AND PHYSIOLOGY FROM THE GREEKS TO HARVEY, Charles 
Singer. Corrected edition of "The Evolution of Anatomy." Classic traces anatomy, phys- 
iology from prescientific times through Greek, Roman periods, dark ages, Renaissance, to 
beginning of modern concepts. Centers on individuals, movements, that definitely advanced 
anatomical knowledge. Plato, Diodes, Erasistratus, Galen, da Vinci, etc. Special section 
on Vesalius. 20 plates. 270 extremely interesting illustrations of ancient, Medieval, Renais- 
sance, Oriental origin, xii + 209pp. 5% x 8. T389 Paperbound $1.75 

SPACE -TIME -MATTER, Hermann Weyl. "The standard treatise on the general theory of 
relativity," (Nature), by world renowned scientist. Deep, clear discussion of logical coher- 
ence of general theory, introducing all needed tools: Maxwell, analytical geometry, non- 
Euclidean geometry, tensor calculus, etc. Basis is classical space-time, before absorption 
of relativity. Contents: Euclidean space, mathematical form, metrical continuum, general 
theory, etc. 15 diagrams, xviii + 330pp. 5% x 8. S267 Paperbound $1.75 



DOVER SCIENCE BOOKS 

MATTER AND MOTION, James Clerk Maxwell. Excellent exposition begins with simple par- 
ticles, proceeds gradually to physical systems beyond complete analysis; motion, force 
properties of centre of mass of material system; work, energy, gravitation, etc. Written 
with all Maxwell's original insights and clarity. Notes by E. Larmor. 17 diagrams. 178pp. 
5% x 8 - S188 Paperbound $1.25 

PRINCIPLES OF MECHANICS, Heinrich Hertz. Last work by the great 19th century physicist 
is not only a classic, but of great interest in the logic of science. Creating a new system 
of mechanics based upon space, time, and mass, it returns to axiomatic analysis, under- 
standing of the formal or structural aspects of science, taking into account logic, observa- 
tion, a priori elements. Of great historical importance to Poincare, Carnap, Einstein, Milne 
A 20 page introduction by R. S. Cohen, Wesleyan University, analyzes the implications of 
gerte s thought and the logic of science. 13 page introduction by Helmholtz. xlii + 274pp 
5% x 8 - S316 Clothbound $3.50 

S317 Paperbound $1.75 

FROM MAGIC TO SCIENCE, Charles Singer. A great historian examines aspects of science 
from Roman Empire through Renaissance. Includes perhaps best discussion of early herbals 
penetrating physiological interpretation of "The Visions of Hildegarde of Bingen." Also 
examines Arabian, Galenic influences; Pythagoras' sphere, Paracelsus; reawakening of 
science under Leonardo da Vinci, Ves^alius; Lorica of Gildas the Briton; etc. Frequent 
quotations with translations from contemporary manuscripts. Unabridged, corrected edi- 
tion. 158 unusual illustrations from Classical, Medieval sources, xxvii + 365pp. 5% x 8. 

T390 Paperbound $2.00 

A HISTORY OF THE CALCULUS, AND ITS CONCEPTUAL DEVELOPMENT, Carl B. Boyer. Provides 
laymen, mathematicians a detailed history of the development of the calculus, from begin- 
nings in antiquity to final elaboration as mathematical abstraction. Gives a sense of 
mathematics not as technique, but as habit of mind, in progression of ideas of Zeno, Plato 
Pythagoras, Eudoxus, Arabic and Scholastic mathematicians, Newton, Leibniz, Taylor, Des- 
cartes, Euler, Lagrange, Cantor, Weierstrass, and others. This first comprehensive, critical 
history of the calculus was originally entitled "The Concepts of the Calculus." Foreword 
by R. Courant. 22 figures. 25 page bibliography, v + 364pp. 5% x 8. 

S509 Paperbound $2.00 

A DIDEROT PICTORIAL ENCYCLOPEDIA OF TRADES AND INDUSTRY, Manufacturing and the 
Technical Arts in Plates Selected from "L'Encyclopfidie ou Dictionnaire Raisonni des 
Sciences, des Arts, et des Metiers" of Denis Diderot. Edited with text by C. Gillispie First 
modern selection of plates from high-point of 18th century French engraving. Storehouse 
of technological information to historian of arts and science. Over 2,000 illustrations on 
485 full page plates, most of them original size, show trades, industries of fascinating 
era in such great detail that modern reconstructions might be made of them. Plates teem 
with men, women, children performing thousands of operations; show sequence general 
operations, closeups, details of machinery. Illustrates such important, interesting trades 
industries as sowing, harvesting, beekeeping, tobacco processing, fishing, arts of war' 
mining, smelting, casting iron, extracting mercury, making gunpowder, cannons, bells' 
shoeing horses, tanning, papermaking, printing, dying, over 45 more categories. Professor 
Gillispie of Princeton supplies full commentary on all plates, identifies operations, tools 
processes, etc. Material is presented in lively, lucid fashion. Of great interest to all 
studying history of science, technology. Heavy library cloth. 920pp. 9 x 12. 

T421 2 volume set $18.50 

DE MAGNETE, William Gilbert. Classic work on magnetism, founded new science. Gilbert 
was first to use word "electricity," to recognize mass as distinct from weight, to discover 
effect of heat on magnetic bodies; invented an electroscope, differentiated between static 
electricity and magnetism, conceived of earth as magnet. This lively work, by first great 
experimental scientist, is not only a valuable historical landmark, but a delightfully easy 
to follow record of a searching, ingenious mind. Translated by P. F. Mottelay. 25 page 
biographical memoir. 90 figures, lix + 368pp. 5% x 8. S470 Paperbound $2.00 

HISTORY OF MATHEMATICS, D. E. Smith. Most comprehensive, non-technical history of math 
in English. Discusses lives and works of over a thousand major, minor figures, with foot- 
notes giving technical information outside book's scheme, and indicating disputed matters 
Vol. I: A chronological examination, from primitive concepts through Egypt Babylonia 
Greece, the Orient, Rome, the Middle Ages, The Renaissance, and to 1900. Vol. II- The 
development of ideas in specific fields and problems, up through elementary calculus. 
"Marks an epoch . . . will modify the entire teaching of the history of science," George 
Sarton. 2 volumes, total of 510 illustrations, 1355pp. 5% x 8. Set boxed in attractive 
container. T429, 430 Paperbound, the set $5.00 

THE PHILOSOPHY OF SPACE AND TIME, H. Reichenbach. An important landmark in develop- 
ment of empiricist conception of geometry, covering foundations of geometry, time theory, 
consequences of Einstein's relativity, including: relations between theory and observations- 
coordinate definitions; relations between topological and metrical properties of space; 
psychological problem of visual intuition of non-Euclidean structures; many more topics 
important to modern science and philosophy. Majority of ideas require only knowledge of 
intermediate math. "Still the best book in the field," Rudolf Carnap. Introduction by 
R. Carnap. 49 figures, xviii + 296pp. 5% x 8. S443 Paperbound $2.00 



CATALOGUE OF 

FOUNDATIONS OF SCIENCE: THE PHILOSOPHY OF THEORY AND EXPERIMENT, N. Campbell. 

A critique of the most fundamental concepts of science, particularly physics. Examines why 
certain propositions are accepted without question, demarcates science from philosophy, 
etc. Part I analyzes presuppositions of scientific thought: existence of material world, 
nature of laws, probability, etc; part 2 covers nature of experiment and applications of 
mathematics: conditions for measurement, relations between numerical laws and theories, 
error, etc. An appendix covers problems arising from relativity, force, motion, space, 
time. A classic in its field. "A real grasp of what science is," Higher Educational Journal, 
xiii + 565pp. 5% x 83/fe. S372 Paperbound $2.95 

THE STUDY OF THE HISTORY OF MATHEMATICS and THE STUDY OF THE HISTORY OF SCIENCE, 
G. Sarton. Excellent introductions, orientation, for beginning or mature worker. Describes 
duty of mathematical historian, incessant efforts and genius of previous generations. Ex- 
plains how today's discipline differs from previous methods. 200 item bibliography with 
critical evaluations, best available biographies of modern mathematicians, best treatises 
on historical methods is especially valuable. 10 illustrations. 2 volumes bound as one. 
113pp. + 75pp. 5% x 8. T240 Paperbound $1.25 

MATHEMATICAL PUZZLES 

MATHEMATICAL PUZZLES OF SAM LOYD, selected and edited by Martin Gardner. 117 choice 
puzzles by greatest American puzzle creator and innovator, from his famous "Cyclopedia 
of Puzzles." All unique style, historical flavor of originals. Based on arithmetic, algebra, 
probability, game theory, route tracing, topology, sliding block, operations research, geo- 
metrical dissection. Includes famous "14-15" puzzle which was national craze, "Horse of 
a Different Color" which sold millions of copies. 120 line drawings, diagrams. Solutions. 
xx + 167pp. 5% x 8. T498 Paperbound $1.00 

SYMBOLIC LOGIC and THE GAME OF LOGIC, Lewis Carroll. "Symbolic Logic" is not concerned 
with modern symbolic logic, but is instead a collection of over 380 problems posed with 
charm and imagination, using the syllogism, and a fascinating diagrammatic method of 
drawing conclusions. In "The Game of Logic" Carroll's whimsical imagination devises a 
logical game played with 2 diagrams and counters (included) to manipulate hundreds of 
tricky syllogisms. The final section, "Hit or Miss" is a lagniappe of 101 additional puzzles 
in the delightful Carroll manner. Until this reprint edition, both of these books were rarities 
costing up to $15 each. Symbolic Logic: Index, xxxi + 199pp. The Game of Logic: 96pp. 

2 vols, bound as one. 5% x 8. T492 Paperbound $1.50 

PILLOW PROBLEMS and A TANGLED TALE, Lewis Carroll. One of the rarest of all Carroll's 
works, "Pillow Problems" contains 72 original math puzzles, all typically ingenious. Particu- 
larly fascinating are Carroll's answers which remain exactly as he thought them out, 
reflecting his actual mental process. The problems in "A Tangled Tale" are in story form, 
originally appearing as a monthly magazine serial. Carroll not only gives the solutions, but 
uses answers sent in by readers to discuss wrong approaches and misleading paths, and 
grades them for insight. Both of these books were rarities until this edition, "Pillow 
Problems" costing up to $25, and "A Tangled Tale" $15. Pillow Problems: Preface and 
Introduction by Lewis Carroll, xx + 109pp. A Tangled Tale: 6 illustrations. 152pp. Two vols, 
bound as one. 5% x 8. T493 Paperbound $1.50 

NEW WORD PUZZLES, G. L. Kaufman. 100 brand new challenging puzzles on words, com- 
binations, never before published. Most are new types invented by author, for beginners 
and experts both. Squares of letters follow chess moves to build words; symmetrical 
designs made of synonyms; rhymed crostics; double word squares; syllable puzzles where 
you fill in missing syllables instead of missing letter; many other types, all new. Solutions. 
"Excellent," Recreation. 100 puzzles. 196 figures, vi + 122pp. 5% x 8. 

T344 Paperbound $1.00 

MATHEMATICAL EXCURSIONS, H. A. Merrill. Fun, recreation, insights into elementary prob- 
lem solving. Math expert guides you on by-paths not generally travelled in elementary math 
courses — divide by inspection, Russian peasant multiplication; memory systems for pi; odd, 
even magic squares; dyadic systems; square roots by geometry; Tchebichev's machine; 
dozens more. Solutions to more difficult ones. "Brain stirring stuff ... a classic," Genie. 
50 illustrations. 145pp. 5% x 8. T350 Paperbound $1.00 

THE BOOK OF MODERN PUZZLES, G. L. Kaufman. Over 150 puzzles, absolutely all new mate- 
rial based on same appeal as crosswords, deduction puzzles, but with different principles, 
techniques. 2-minute teasers, word labyrinths, design, pattern, logic, observation puzzles, 
puzzles testing ability to apply general knowledge to peculiar situations, many others. 
Solutions. 116 illustrations. 192pp. 5% x 8. T143 Paperbound $1.00 

MATHEMAGIC, MAGIC PUZZLES, AND GAMES WITH NUMBERS, R. V. Heath. Over 60 puzzles, 
stunts, on properties of numbers. Easy techniques for multiplying large numbers mentally, 
identifying unknown numbers, finding date of any day in any year. Includes The Lost Digit, 

3 Acrobats, Psychic Bridge, magic squares, trjangles, cubes, others not easily found else- 
where. Edited by J. S. Meyer. 76 illustrations. 128pp. 5% x 8. T110 Paperbound $1.00 



DOVER SCIENCE BOOKS 

PUZZLE QUIZ AND STUNT FUN, J. Meyer. 238 high-priority puzzles, stunts, tricks— math 
puzzles like The Clever Carpenter, Atom Bomb, Please Help Alice; mysteries, deductions 
like The Bridge of Sighs, Secret Code; observation puzzlers like The American Flag, Playing 
Cards, Telephone Dial; over 200 others with magic squares, tongue twisters, puns, ana- 
grams. Solutions. Revised, enlarged edition of "Fun-To-Do." Over 100 illustrations. 238 
puzzles, stunts, tricks. 256pp. 5% x 8. T337 Paperbound $1.00 

101 PUZZLES IN THOUGHT AND LOGIC, C. R. Wylie, Jr. For readers who enjoy challenge, 
stimulation of logical puzzles without specialized math or scientific knowledge. Problems 
entirely new, range from relatively easy to brainteasers for hours of subtle entertainment. 
Detective puzzles, find the lying fisherman, how a blind man identifies color by logic, many 
more. Easy-to-understand introduction to logic of puzzle solving and general scientific 
method. 128pp. 5% x 8. T367 Paperbound $1.00 

CRYPTANALYSIS, H. F. Gaines. Standard elementary, intermediate text for serious students. 
Not just old material, but much not generally known, except to experts. Concealment, 
Transposition, Substitution ciphers; Vigenere, Kasiski, Playfair, multafid, dozens of other 
techniques. Formerly "Elementary Cryptanalysis." Appendix with sequence charts, letter 
frequencies in English, 5 other languages, English word frequencies. Bibliography. 167 
codes. New to this edition: solutions to codes, vi + 230pp. 5% x 8%. 

T97 Paperbound $1.95 

CRYPTOGRAPY, L. D. Smith. Excellent elementary introduction to enciphering, deciphering 
secret writing. Explains transposition, substitution ciphers; codes; solutions; geometrical 
patterns, route transcription, columnar transposition, other methods. Mixed cipher systems; 
single, polyalphabetical substitutions; mechanical devices; Vigenere; etc. Enciphering Jap- 
anese; explanation of Baconian biliteral cipher; frequency tables. Over 150 problems. Bib- 
liography. Index. 164pp. 5% x 8. T247 Paperbound $1.00 

MATHEMATICS, MAGIC AND MYSTERY, M. Gardner. Card tricks, metal mathematics, stage 
mind-reading, other "magic" explained as applications of probability, sets, number theory, 
etc. Creative examination of laws, applications. Scores of new tricks, insights. 115 sections 
on cards, dice, coins; vanishing tricks, many others. No sleight of hand — math guarantees 
success. "Could hardly get more entertainment . . . easy to follow," Mathematics Teacher. 
115 illustrations, xii + 174pp. 5% x 8. T335 Paperbound $1.00 

AMUSEMENTS IN MATHEMATICS, H. E. Dudeney. Foremost British originator of math puzzles, 
always witty, intriguing, paradoxical in this classic. One of largest collections. More than 
430 puzzles, problems, paradoxes. Mazes, games, problems on number manipulations, 
unicursal, other route problems, puzzles on measuring, weighing, packing, age, kinship, 
chessboards, joiners', crossing river, plane figure dissection, many others. Solutions. More 
than 450 illustrations, viii + 258pp. 5% x 8. T473 Paperbound $1.25 

THE CANTERBURY PUZZLES H. E. Dudeney. Chaucer's pilgrims set one another problems in 
story form. Also Adventures of the Puzzle Club, the Strange Escape of the King's Jester, 
the Monks of Riddlewell, the Squire's Christmas Puzzle Party, others. All puzzles are 
original, based on dissecting plane figures, arithmetic, algebra, elementary calculus, other 
branches of mathematics, and purely logical ingenuity. "The limit of ingenuity and in- 
tricacy," The Observer. Over 110 puzzles, full solutions. 150 illustrations, viii + 225 pp. 
5% x 8. T474 Paperbound $1.25 

MATHEMATICAL PUZZLES FOR BEGINNERS AND ENTHUSIASTS, G. Mott-Smith. 188 puzzles to 
test mental agility. Inference, interpretation, algebra, dissection of plane figures, geometry, 
properties of numbers, decimation, permutations, probability, all are in these delightful 
problems. Includes the Odic Force, How to Draw an Ellipse, Spider's Cousin, more than 180 
others. Detailed solutions. Appendix with square roots, triangular numbers, primes, etc. 
135 illustrations. 2nd revised edition. 248pp. 5% x 8. T198 Paperbound $1.00 

MATHEMATICAL RECREATIONS, M. Kraitchik. Some 250 puzzles, problems, demonstrations of 
recreation mathematics on relatively advanced level. Unusual historical problems from 
Greek, Medieval, Arabic, Hindu sources; modem problems on "mathematics without num- 
bers," geometry, topology, arithmetic, etc. Pastimes derived from figurative, Mersenne, 
Fermat numbers: fairy chess; latruncles: reversi; etc. Full solutions. Excellent insights 
into special fields of math. "Strongly recommended to all who are interested in the 
lighter side of mathematics," Mathematical Gaz. 181 illustrations. 330pp. 5% x 8. 

T163 Paperbound $1.75 

FICTION 

FLATLAND, E. A. Abbott. A perennially popular science-fiction classic about life in a 2- 
dimensional world, and the impingement of higher dimensions. Political, satiric, humorous, 
moral overtones. This land where women are straight lines and the lowest and most dan- 
gerous classes are isosceles triangles with 3° vertices conveys brilliantly a feeling for 
many concepts of modern science. 7th edition. New introduction by Banesh Hoffmann. 128pp. 
5% x 8 Tl Paperbound $1.00 



CATALOGUE OF 

SEVEN SCIENCE FICTION NOVELS OF H. 6. WELLS. Complete texts, unabridged, of seven of 
Wells' greatest novels: The War of the Worlds, The Invisible Man, The Island of Or. Moreau, 
The Food of the Gods, First Men in the Moon, In the Days of the Comet, The Time Machine. 
Still considered by many experts to be the best science-fiction ever written, they will offer 
amusements and instruction to the scientific minded reader. "The great master," Sky and 
Telescope. 1051pp. 5% x 8. T264 Clothbound $3.95 

28 SCIENCE FICTION STORIES OF H. 6. WELLS. Unabridged! This enormous omnibus contains 
2 full length novels — Men Like Gods, Star Begotten — plus 26 short stories of space, time, 
invention, biology, etc. The Crystal Egg, The Country of the Blind, Empire of the Ants, 
The Man Who Could Work Miracles, Aepyornis Island, A Story of the Days to Come, and 
20 others "A master ... not surpassed by . . . writers of today," The English Journal. 
915pp. 5% x 8. T265 Clothbound $3.95 

FIVE ADVENTURE NOVELS OF H. RIDER HAGGARD. All the mystery and adventure of darkest 
Africa captured accurately by a man who lived among Zulus for years, who knew African 
ethnology, folkways as did few of his contemporaries. They have been regarded as examples 
of the very best high adventure by such critics as Orwell, Andrew Lang, Kipling. Contents: 
She, King Solomon's Mines, Allan Quatermain, Allan's Wife, Maiwa's Revenge. "Could spin 
a yarn so full of suspense and color that you couldn't put the story down," Sat. Review. 
821pp. 5% x 8. T108 Clothbound $3.95 



CHESS AND CHECKERS 

LEARN CHESS FROM THE MASTERS, Fred Reinfeld. Easiest, most instructive way to im- 
prove your game — play 10 games against such masters as Marshall, Znosko-Borovsky, Bron- 
stein, Najdorf, etc., with each move graded by easy system. Includes ratings for alternate 
moves possible. Games selected for interest, clarity, easily isolated principles. Covers 
Ruy Lopez, Dutch Defense, Vienna Game openings; subtle, intricate middle game variations; 
all-important end game. Full annotations. Formerly "Chess by Yourself." 91 diagrams, viii 
+ 144pp. 5% x 8. T362 Paperbound $1.00 

REINFELD ON THE END GAME IN CHESS, Fred Reinfeld. Analyzes 62 end games by Alekhine, 
Flohr, Tarrasch, Morphy, Capablanca, Rubinstein, Lasker, Reshevsky, other masters. Only 
1st rate book with extensive coverage of error— tell exactly what is wrong with each move 
you might have made. Centers around transitions from middle play to end play. King and 
pawn, minor pieces, queen endings; blockage, weak, passed pawns, etc. "Excellent ... a 
boon," Chess Life. Formerly "Practical End Play." 62 figures, vi + 177pp. 5% x 8. 

T417 Paperbound $1.25 

HYPERMODERN CHESS as developed in the games of its greatest exponent, ARON NIMZO- 
VICH, edited by Fred Reinfeld. An intensely original player, analyst, Nimzovich's approaches 
startled, often angered the chess world. This volume, designed for the average player, 
shows how his iconoclastic methods won him victories over Alekhine, Lasker, Marshall, 
Rubinstein, Spielmann, others, and infused new life into the game. Use his methods to 
startle opponents, invigorate play. "Annotations and introductions to each game ... are 
excellent," Times (London). 180 diagrams, viii + 220pp. 5% x 8. T448 Paperbound $1.35 

THE ADVENTURE OF CHESS, Edward Lasker. Lively reader, by one of America's finest chess 
masters, including: history of chess, from ancient Indian 4-handed game of Chaturanga 
to great players of today; such delights and oddities as Maelzel's chess-playing automaton 
that beat Napoleon 3 times; etc. One of most valuable features is author's personal recollec- 
tions of men he has played against — Nimzovich, Emanuel Lasker, Capablanca, Alekhine, 
etc. Discussion of chess-playing machines (newly revised). 5 page chess primer. 11 illus- 
trations. 53 diagrams. 296pp. 5% x 8. S510 Paperbound $1.45 

THE ART OF CHESS, James Mason. Unabridged reprinting of latest revised edition of most 
famous general study ever written. Mason, early 20th century master, teaches beginning, 
intermediate player over 90 openings; middle game, end game, to see more moves ahead, 
to plan purposefully, attack, sacrifice, defend, exchange, govern general strategy. "Classic 
... one of the clearest and best developed studies," Publishers Weekly. Also included, a 
complete supplement by F. Reinfeld, "How Do You Play Chess?", invaluable to beginners 
for its lively question-and-answer method. 448 diagrams. 1947 Reinfeld-Bernstein text. 
Bibliography, xvi + 340pp. 5% x 8. T463 Paperbound $1.85 

MORPHY'S GAMES OF CHESS, edited by P. W. Sergeant. Put boldness into your game by 
flowing brilliant, forceful moves of the greatest chess player of all time. 300 of Morphy's 
best games, carefully annotated to reveal principles. 54 classics against masters like 
Anderssen, Harrwitz, Bird, Paulsen, and others. 52 games at odds; 54 blindfold games; plus 
over 100 others. Follow his interpretation of Dutch Defense, Evans Gambit, Giuoco Piano, 
Ruy Lopez, many more. Unabridged reissue of latest revised edition. New introduction by 
F. Reinfeld. Annotations, introduction by Sergeant. 235 diagrams, x + 352pp. 5% x 8. 

T386 Paperbound $1.75 

8 



DOVER SCIENCE BOOKS 

WIN AT CHECKERS, M. Hopper. (Formerly "Checkers.") Former World's Unrestricted Checker 
Champion discusses principles of game, expert's shots, traps, problems for beginner, stand- 
ard openings, locating best move, end game, opening "blitzkrieg" moves to draw when 
behind, etc. Over 100 detailed questions, answers anticipate problems. Appendix. 75 prob- 
lems with solutions, diagrams. 79 figures, xi + 107pp. 5% x 8. T363 Paperbound $1.00 

HOW TO FORCE CHECKMATE, Fred Reinfeld. If you have trouble finishing off your opponent, 
here is a collection of lightning strokes and combinations from actual tournament play. 
Starts with 1-move checkmates, works up to 3-move mates. Develops ability to lock ahead, 
gain new insights into combinations, complex or deceptive positions,- ways to estimate weak- 
nesses, strengths of you and your opponent. "A good deal of amusement and instruction," 
Times, (London). 300 diagrams. Solutions to all positions. Formerly "Challenge to Chess 
Players." 111pp. 5% x 8. T417 Paperbound |1.25 

A TREASURY OF CHESS LORE, edited by Fred Reinfeld. Delightful collection of anecdotes, 
short stories, aphorisms by, about masters-, poems, accounts of games, tournaments, photo- 
graphs; hundreds of humorous, pithy, satirical, wise, historical episodes, comments, word 
portraits. Fascinating "must" for chess players; revealing and perhaps seductive to those 
who wonder what their friends see in game. 49 photographs (14 full page plates). 12 
diagrams, xi + 306pp. 5% x 8. T458 Paperbound SI. 75 

WIN AT CHESS, Fred Reinfeld. 300 practical chess situations, to sharpen your eye, test skill 
against masters. Start with simple examples, progress at own pace to complexities. This 
selected series of crucial moments in chess will stimulate imagination, develop stronger, 
more versatile game. Simple grading system enables you to judge progress. "Extensive use 
of diagrams is a great attraction," Chess. 300 diagrams. Notes, solutions to every situation. 
Formerly "Chess Quiz." vi + 120pp. 5% x 8. T433 Paperbound $1.00 



MATHEMATICS: 

ELEMENTARY TO INTERMEDIATE 

HOW TO CALCULATE QUICKLY, H. Sticker. Tried and true method to help mathematics of 
everyday life. Awakens "number sense"— ability to see relationships between numbers as 
whole quantities. A serious course of over 9000 problems and their solutions through 
techniques not taught in schools: left-to-right multiplications, new fast division, etc. 10 
minutes a day will double or triple calculation speed. Excellent for scientist at home in 
higher math, but dissatisfied with speed and accuracy in lower math. 256pp. 5 x 7V*. 

Paperbound $1.00 

FAMOUS PROBLEMS OF ELEMENTARY GEOMETRY, Felix Klein. Expanded version of 1894 
Easter lectures at Gottingen. 3 problems of classical geometry: squaring the circle, trisect- 
ing angle, doubling cube, considered with full modern implications: transcendental num- 
bers, pi, etc. "A modern classic ... no knowledge of higher mathematics is required," 
Scientia. Notes by R. Archibald. 16 figures, xi + 92pp. 5% x 8. T298 Paperbound $1.00 

HIGHER MATHEMATICS FOR STUDENTS OF CHEMISTRY ANO PHYSICS, J. W. Mellor. Practical, 
not abstract, building problems out of familiar laboratory material. Covers differential cal- 
culus, coordinate, analytical geometry, functions, integral calculus, infinite series, numerical 
equations, differential equations, Fourier's theorem probability, theory of errors, calculus 
of variations, determinants. "If the reader is not familiar with this book, it will repay 
him to examine it," Chem. and Engineering News. 800 problems. 189 figures, xxi + 641pp. 
5% x 8. S193 Paperbound $2.25 

TRIGONOMETRY REFRESHER FOR TECHNICAL MEN, A. A. Klaf. 913 detailed questions, answers 
cover most important aspects of plane, spherical trigonometry — particularly useful in clearing 
up difficulties in special areas. Part I: plane trig, angles, quadrants, functions, graphical repre- 
sentation, interpolation, equations, logs, solution of triangle, use of slide rule, etc. Next 
188 pages discuss applications to navigation, surveying, elasticity, architecture, other 
special fields. Part 3: spherical trig, applications to terrestrial, astronomical problems. 
Methods of time-saving, simplification of principal angles, make book most useful. 913 
questions answered. 1738 problems, answers to odd numbers. 494 figures. 24 pages of for- 
mulas, functions, x + 629pp. 5% x 8. T371 Paperbound $2.00 

CALCULUS REFRESHER FOR TECHNICAL MEN, A. A. Klaf. 756 questions examine most im- 
portant aspects of integral, differential calculus. Part I: simple differential calculus, con- 
stants, variables, functions, increments, logs, curves, etc. Part 2: fundamental ideas of 
integrations, inspection, substitution, areas, volumes, mean value, double, triple integration, 
etc. Practical aspects stressed. 50 pages illustrate applications to specific problems of civil, 
nautical engineering, electricity, stress, strain, elasticity, similar fields. 756 questions 
answered. 566 problems, mostly answered. 36pp. of useful constants, formulas, v + 431pp. 
5% x 8. T370 Paperbound $2.00 



CATALOGUE OF 

MONOGRAPHS ON TOPICS OF MODERN MATHEMATICS, edited by J. W. A. Young. Advanced 
mathematics for persons who have forgotten, or not gone beyond, high school algebra 
9 monographs on foundation of geometry, modern pure geometry, non-Euclidean geometry, 
fundamental propositions of algebra, algebraic equations, functions, calculus, theory of 
numbers, etc. Each monograph gives proofs of important results, and descriptions of lead- 
ing methods, to provide wide coverage. "Of high merit," Scientific American. New intro- 
duction by Prof. M. Kline, N.Y. Univ. 100 diagrams, xvi + 416pp. 6Vs x 9V4. 

S289 Paperbound $2.00 

MATHEMATICS IN ACTION, 0. G. Sutton. Excellent middle level application of mathematics 
to study of universe, demonstrates how math is applied to ballistics, theory of computing 
machines, waves, wave-like phenomena, theory of fluid flow, meteorological problems, 
statistics, flight, similar phenomena. No knowledge of advanced math required. Differential 
equations, Fourier series, group concepts, Eigentunctions, Planck's constant, airfoil theory, 
and similar topics explained so clearly in everyday language that almost anyone can derive 
benefit from reading this even if much of high-school math is forgotten. 2nd edition. 88 
figures, viii + 236pp. 5% x 8. T450 Clothbound $3.50 

ELEMENTARY MATHEMATICS FROM AN ADVANCED STANDPOINT, Felix Klein. Classic text, 
an outgrowth of Klein's famous integration and survey course at Gottingen. Using one field 
to interpret, adjust another, it covers basic topics in each area, with extensive analysis. 
Especially valuable in areas of modern mathematics. "A great mathematician, inspiring 
teacher, . . . deep insight," Bui., Amer. Math Soc. 

Vol. I. ARITHMETIC, ALGEBRA, ANALYSIS. Introduces concept of function immediately, en- 
livens discussion with graphical, geometric methods. Partial contents: natural numbers, 
special properties, complex numbers. Real equations with real unknowns, complex quan- 
tities. Logarithmic, exponential functions, infinitesimal calculus. Transcendence of e and pi, 
theory of assemblages. Index. 125 figures, ix + 274pp. 5% x 8. S151 Paperbound $1.75 

Vol. II. GEOMETRY. Comprehensive view, accompanies space perception inherent in geom- 
etry with analytic formulas which facilitate precise formulation. Partial contents: Simplest 
geometric manifold; line segments, Grassman determinant principles, classication of con- 
figurations of space. Geometric transformations: affine, projective, higher point transforma- 
tions, theory of the imaginary. Systematic discussion of geometry and its foundations. 141 
illustrations, ix + 214pp. 5% x 8. S151 Paperbound $1.75 

A TREATISE ON PLANE AND ADVANCED TRIGONOMETRY, E. W. Hobson. Extraordinarily wide 
coverage, going beyond usual college level, one of few works covering advanced trig in 
full detail. By a great expositor with unerring anticipation of potentially difficult points. 
Includes circular functions; expansion of functions of multiple angle; trig tables; relations 
between sides, angles of triangles; complex numbers; etc. Many problems fully solved. 
"The best work on the subject," Nature. Formerly entitled "A Treatise on Plane Trigonom- 
etry." 689 examples. 66 figures, xvi + 383pp. 5% x 8. S353 Paperbound $1.95 

NON-EUCLIDEAN GEOMETRY, Roberto Bonola. The standard coverage of non-Euclidean geom- 
etry. Examines from both a historical and mathematical point of view geometries which 
have arisen from a study of Euclid's 5th postulate on parallel lines. Also included are 
complete texts, translated, of Bolyai's "Theory of Absolute Space," Lobachevsky's "Theory 
of Parallels." 180 diagrams. 431pp. 5% x 8. S27 Paperbound $1.95 

GEOMETRY OF FOUR DIMENSIONS, H. P. Manning. Unique in English as a clear, concise intro- 
duction. Treatment is synthetic, mostly Euclidean, though in hyperplanes and hyperspheres 
at infinity, non-Euclidean geometry is used. Historical introduction. Foundations of 4-dimen- 
sional geometry. Perpendicularity, simple angles. Angles of planes, higher order. Symmetry, 
order, motion; hyperpyramids, hypercones, hyperspheres; figures with parallel elements; 
volume, hypervolume in space; regular polyhedroids. Glossary. 78 figures, ix + 348pp. 
5 % x 8 - S182 Paperbound $1.95 



MATHEMATICS: INTERMEDIATE TO ADVANCED 
GEOMETRY (EUCLIDEAN AND NON-EUCLIDEAN) 

THE GEOMETRY OF RENE' DESCARTES. With this book, Descartes founded analytical geometry. 
Original French text, with Descartes's own diagrams, and excellent Smith-Latham transla- 
tion. Contains: Problems the Construction of Which Requires only Straight Lines and Circles; 
On the Nature of Curved Lines; On the Construction of Solid or Supersolid Problems. Dia- 
grams. 258pp. 5% x 8. S68 Paperbound $1.50 

10 



DOVER SCIENCE BOOKS 

THE WORKS OF ARCHIMEDES, edited by T. L. Heath. Ail the known works of the great Greek 
mathematician, including the recently discovered Method of Archimedes. Contains: On 
Sphere and Cylinder, Measurement of a Circle, Spirals, Conoids, Spheroids, etc. Definitive 
edition of greatest mathematical intellect of ancient world. 186 page study by Heath dis- 
cusses Archimedes and history of Greek mathematics. 563pp. 5% x 8. S9 Paperbound $2.00 

COLLECTED WORKS OF BERNARD RIEMANN. Important sourcebook, first to contain complete 
text of 1892 "Werke" and the 1902 supplement, unabridged. 31 monographs, 3 complete 
lecture courses, 15 miscellaneous papers which have been of enormous importance in 
relativity, topology, theory of complex variables, other areas of mathematics. Edited by 
R. Dedekind, H. Weber, M. Noether, W. Wirtinger. German text; English introduction by 
Hans Lewy. 690pp. 5% x 8. S226 Paperbound $2.85 

THE THIRTEEN BOOKS OF EUCLID'S ELEMENTS, edited by Sir Thomas Heath. Definitive edition 
of one of very greatest classics of Western world. Complete translation of Heiberg text, 
plus spurious Book XIV. 150 page introduction on Greek, Medieval mathematics, Euclid, 
texts, commentators, etc. Elaborate critical apparatus parallels text, analyzing each defini- 
tion, postulate, proposition, covering textual matters, refutations, supports, extrapolations, 
etc. This is the full Euclid. Unabridged reproduction of Cambridge U. 2nd edition. 3 vol- 
umes. 995 figures. 1426pp. 5% x 8. S88, 89, 90, 3 volume set, paperbound $6.00 

AN INTRODUCTION TO GEOMETRY OF N DIMENSIONS, D. M. Y. Sommerville. Presupposes no 
previous knowledge of field. Only book in English devoted exclusively to higher dimensional 
geometry. Discusses fundamental ideas of incidence, parallelism, perpendicularity, angles 
between linear space, enumerative geometry, analytical geometry from projective and metric 
views, polytopes, elementary ideas in analysis situs, content of hyperspacial figures. 60 
diagrams. 196pp. 5% x 8. S494 Paperbound $1.50 

ELEMENTS OF NON-EUCLIDEAN GEOMETRY, D. M. Y. Sommerville. Unique in proceeding step- 
by-step. Requires only good knowledge of high-school geometry and algebra, to grasp ele- 
mentary hyperbolic, elliptic, analytic non-Euclidean Geometries; space curvature and its 
implications; radical axes; homopethic centres and systems of circles; parataxy and parallel- 
ism; Gauss' proof of defect area theorem; much more, with exceptional clarity. 126 prob- 
lems at chapter ends. 133 figures, xvi + 274pp. 5% x 8. S460 Paperbound $1.50 

THE FOUNDATIONS OF EUCLIDEAN GEOMETRY, H. G. Forder. First connected, rigorous ac- 
count in light of modern analysis, establishing propositions without recourse to empiricism, 
without multiplying hypotheses. Based on tools of 19th and 20th century mathematicians, 
who made it possible to remedy gaps and complexities, recognize problems not earlier 
discerned. Begins with important relationship of number systems in geometrical figures. 
Considers classes, relations, linear order, natural numbers, axioms for magnitudes, groups, 
quasi-flelds, fields, non-Archimedian systems, the axiom system (at length), particular axioms 
(two chapters on the Parallel Axioms), constructions, congruence, similarity, etc. Lists.- 
axioms employed, constructions, symbols in frequent use. 295pp. 53/% x 8. 

S481 Paperbound $2.00 

CALCULUS, FUNCTION THEORY (REAL AND COMPLEX), 
FOURIER THEORY 

FIVE VOLUME "THEORY OF FUNCTIONS" SET BY KONRAD KNOPP. Provides complete, readily 
followed account of theory of functions. Proofs given concisely, yet without sacrifice of 
completeness or rigor. These volumes used as texts by such universities as M.I.T., Chicago, 
N.Y. City College, many others. "Excellent introduction . . . remarkably readable, concise, 
clear, rigorous," J. of the American Statistical Association. 

ELEMENTS OF THE THEORY OF FUNCTIONS, Konrad Knopp. Provides background for further 
volumes in this set, or texts on similar level. Partial contents: Foundations, system of com- 
plex numbers and Gaussian plane of numbers, Riemann sphere of numbers, mapping by 
linear functions, normal forms, the logarithm, cyclometric functions, binomial series. "Not 
only for the young student, but also for the student who knows all about what is in it," 
Mathematical Journal. 140pp. 5% x 8. S154 Paperbound $1.35 

THEORY OF FUNCTIONS, PART I, Konrad Knopp. With volume II, provides coverage of basic 
concepts and theorems. Partial contents: numbers and points, functions of a complex 
variable, integral of a continuous function, Cauchy's intergral theorem, Cauchy's integral 
formulae, series with variable terms, expansion and analytic function in a power series, 
analytic continuation and complete definition of analytic '-"ictions, Laurent expansion, types 
of singularities, vii + 146pp. 5% x 8. S156 Paperbound $1.35 

THEORY OF FUNCTIONS, PART II, Konrad Knopp. Application and further development of 
general theory, special topics. Single valued functions, entire, Weierstrass. Meromorphic 
functions: Mittag-Leffler. Periodic functions. Multiple valued functions. Riemann surfaces. 
Algebraic functions. Analytical configurations, Riemann surface, x + 150pp. 5% x 8. 

S157 Paperbound $1.35 

11 



CATALOGUE OF 

PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME I, Konrad Knopp. Problems in ele- 
mentary theory, for use with Knopp's "Theory of Functions," or any other text. Arranged 
according to increasing difficulty. Fundamental concepts, sequences of numbers and infinite 
series, complex variable, integral theorems, development in series, conformal mapping. 
Answers, viii + 126pp. 5% x 8. S 158 Paperbound $1.35 

PROBLEM BOOK IN THE THEORY OF FUNCTIONS, VOLUME II, Konrad Knopp. Advanced theory 
of functions, to be used with Knopp's "Theory of Functions," or comparable text. Singular- 
ities, entire and meromorphic functions, periodic, analytic, continuation, multiple-valued 
functions, Riemann surfaces, conformal mapping. Includes section of elementary problems 
"The difficult task of selecting . . . problems just within the reach of the beginner is 
here masterfully accomplished," AM. MATH. SOC. Answers. 138pp. 5*/b x 8. 

S159 Paperbound $1.35 

ADVANCED CALCULUS, E. B. Wilson. Still recognized as one of most comprehensive, useful 
texts. Immense amount of well-represented, fundamental material, including chapters on 
vector functions, ordinary differential equations, special functions, calculus of variations 
etc., which are excellent introductions to these areas. Requires only one year of calculus' 
Over 1300 exercises cover both pure math and applications to engineering and physical 
problems. Ideal reference, refresher. 54 page introductory review, ix + 566pp. 5% x 8 

S504 Paperbound $2.45 

LECTURES ON THE THEORY OF ELLIPTIC FUNCTIONS, H. Hancock. Reissue of only book in 
English with so extensive a coverage, especially of Abel, Jacobi, Legendre, Weierstrass 
Hermite, Liouville, and Riemann. Unusual fullness of treatment, plus applications as well as 
theory in discussing universe of elliptic integrals, originating in works of Abel and 
Jacobi. Use is made of Riemann to provide most general theory. 40-page table of formulas 
76 figures, xxiii + 498pp. 5% x 8. S483 Paperbound $2.55 

THEORY OF FUNCTIONALS AND OF INTEGRAL AND INTEGRO-DIFFERENTIAL EQUATIONS, Vito 
Volterra. Unabridged republication of only English translation, General theory of functions 
depending on continuous set of values of another function. Based on author's concept of 
transition from finite number of variables to a continually infinite number. Includes much 
material on calculus of variations. Begins with fundamentals, examines generalization of 
analytic functions, functional derivative equations, applications, other directions of theory 
etc. New introduction by G. C. Evans. Biography, criticism of Volterra's work by E Whit- 
taker, xxxx + 226pp. 5% x 8. S502 Paperbound $1.75 

AN INTRODUCTION TO FOURIER METHODS AND THE LAPLACE TRANSFORMATION, Philip 
Franklin. Concentrates on essentials, gives broad view, suitable for most applications. Re- 
quires only knowledge of calculus. Covers complex qualities with methods of computing ele- 
mentary functions for complex values of argument and finding approximations by charts; 
Fourier series; harmonic anaylsis; much more. Methods are related to physical problems 
of heat flow, vibrations, electrical transmission, electromagnetic radiation, etc. 828 prob- 
lems, answers. Formerly entitled "Fourier Methods." x + 289pp. 5% x 8. 

S452 Paperbound $1.75 

THE ANALYTICAL THEORY OF HEAT, Joseph Fourier. This book, which revolutionized mathe- 
matical physics, has been used by generations of mathematicians and physicists interested 
in heat or application of Fourier integral. Covers cause and reflection of rays of heat, 
radiant heating, heating of closed spaces, use of trigonometric series in theory of heat, 
Fourier integral, etc. Translated by Alexander Freeman. 20 figures, xxii + 466pp. 5% x 8. 

S93 Paperbound $2.00 

ELLIPTIC INTEGRALS, H. Hancock. Invaluable in work involving differential equations with 
cubics, quatrics under root sign, where elementary calculus methods are inadequate. Prac- 
tical solutions to problems in mathematics, engineering, physics; differential equations re- 
quiring integration of Lam6's, Briot's, or Bouquet's equations; determination of arc of 
ellipse, hyperbola, lemiscate; solutions of problems in elastics; motion of a projectile under 
resistance varying as the cube of the velocity; pendulums; more. Exposition in accordance 
with Legendre-Jacobi theory. Rigorous discussion of Legendre transformations. 20 figures. 
5 place table. 104pp. 5% x 8. S484 Paperbound $1.25 

THE TAYLOR SERIES, AN INTRODUCTION TO THE THEORY OF FUNCTIONS OF A COMPLEX 
VARIABLE, P. Dienes. Uses Taylor series to approach theory of functions, using ordinary 
calculus only, except in last 2 chapters. Starts with introduction to real variable and com- 
plex algebra, derives properties of infinite series, complex differentiation, integration, etc. 
Covers biuniform mapping, overconvergence and gap theorems, Taylor series on its circle 
of convergence, etc. Unabridged corrected reissue of first edition. 186 examples, many 
fully worked out. 67 figures, xii + 555pp. 5% x 8. S391 Paperbound $2.75 

LINEAR INTEGRAL EQUATIONS, W. V. Lovitt. Systematic survey of general theory, with some 
application to differential equations, calculus of variations, problems of math, physics. 
Includes: integral equation of 2nd kind by successive substitutions; Fredholm's equation 
as ratio of 2 integral series in lambda, applications of the Fredholm theory, Hilbert-Schmidt 
theory of symmetric kernels, application, etc. Neumann, Dirichlet, vibratory problems, 
ix + 253pp. 5% x 8. S175 Clothbound $3.50 

S176 Paperbound $1.60 

12 



DOVER SCIENCE BOOKS 

DICTIONARY OF CONFORMAL REPRESENTATIONS, H. Kober. Developed by British Admiralty to 
solve Laplace's equation in 2 dimensions. Scores of geometrical forms and transformations 
for electrical engineers, Joukowski aerofoil for aerodynamics, Schwartz-Christoffel trans- 
formations for hydro-dynamics, transcendental functions. Contents classified according to 
analytical functions describing transformations with corresponding regions. Glossary. Topo- 
logical index. 447 diagrams. 6Va x 9V*. -S160 Paperbound $2.00 

ELEMENTS OF THE THEORY OF REAL FUNCTIONS, J. E. Littlewood. Based on lectures at 
Trinity College, Cambridge, this book has proved extremely successful in introducing graduate 
students to modern theory of functions. Offers full and concise coverage of classes and 
cardinal numbers, well ordered series, other types of series, and elements of the theory 
of sets of points. 3rd revised edition, vii + 71pp. 5% x 8. S171 Clothbound $2.85 

S172 Paperbound $1.25 

INFINITE SEQUENCES AND SERIES, Konrad Knopp. 1st publication in any language. Excellent 
introduction to 2 topics of modern mathematics, designed to give student background to 
penetrate further alone. Sequences and sets, real and complex numbers, etc. Functions of 
a real and complex variable. Sequences and series. Infinite series. Convergent power series. 
Expansion of elementary functions. Numerical evaluation of series, v + 186pp. 5% x 8. 

5152 Clothbound 53.50 

5153 Paperbound $1.75 

THE THEORY AND FUNCTIONS OF A REAL VARIABLE AND THE THEORY OF FOURIER'S SERIES, 
E. W .Hobson. One of the best introductions to set theory and various aspects of functions 
and Fourier's series. Requires only a good background in calculus. Exhaustive .coverage of: 
metric and descriptive properties of sets of points; transfinite numbers and order types; 
functions of a real variable; the Riemann and Lebesgue integrals; sequences and series 
of numbers; power-series; functions representable by series sequences of continuous func- 
tions; trigonometrical series; representation of functions by Fourier's series; and much 
more. "The best possible guide," Nature. Vol. I: 88 detailed examples, 10 figures. Index, 
xv + 736pp. Vol. II: 117 detailed examples, 13 figures, x + 780pp. 6% * t 9V4. _, MM 

Vol. I: S387 Paperbound $3.00 
Vol. II: S388 Paperbound $3.00 

ALMOST PERIODIC FUNCTIONS, A. S. Besicovitch. Unique and important summary by a well 
known mathematician covers in detail the two stages of development in Bohr's theory 
of almost periodic functions: (1) as a generalization of pure periodicity, with results and 
proofs; (2) the work done by Stepanof, Wiener, Weyl, and Bohr in generalizing the theory, 
xi + 180pp. 5% x 8. S18 Paperbound $1.75 

INTRODUCTION TO THE THEORY OF FOURIER'S SERIES AND INTEGRALS, H. S. Carslaw. 3rd 

revised edition, an outgrowth of author's courses at Cambridge. Historical introduction, 
rational, irrational numbers, infinite sequences and series, functions of a single variable, 
definite integral, Fourier series, and similar topics. Appendices discuss practical harmonic 
analysis, periodogram analysis, Lebesgue's theory. 84 examples, xiii + 368pp. 5% x», 

S48 Paperbound $2.00 



SYMBOLIC LOGIC 

THE ELEMENTS OF MATHEMATICAL LOGIC, Paul Rosenbloom. First publication in any lan- 
guage. For mathematically mature readers with no training in symbolic, logic. Development 
of lectures given at Lund Univ., Sweden, 1948. Partial contents: Logic of classes, funda- 
mental theorems, Boolean algebra, logic of propositions, of propositional functions, expres- 
sive languages, combinatory logics, development of math within an object language, para- 
doxes, theorems of Post, Goedel, Church, and similar topics, iv + 214pp. 5% x 8. 

S227 Paperbound $1.45 

INTRODUCTION TO SYMBOLIC LOGIC AND ITS APPLICATION, R. Carnap. Clear, comprehensive, 
rigorous, by perhaps greatest living master. Symbolic languages analyzed, one constructed. 
Applications to math (axiom systems for set theory, real, natural numbers), topology 
(Dedekind, Cantor continuity explanations), physics (general analysis of determination, cau- 
sality, space-time topology), biology (axiom system for basic concepts). "A masterpiece, 
Zentralblatt fur Mathematik und Ihre Grenzgebiete. Over 300 exercises. 5 figures, xvi + 
241pp. 5% x 8. S453 Paperbound $1.85 

AN INTRODUCTION TO SYMBOLIC LOGIC, Susanne K. Langer. Probably clearest book for the 
philosopher, scientist, layman — no special knowledge of math required. Starts with simplest 
symbols, goes on to give remarkable grasp of Boole-Schroeder, Russell-Whitehead systems, 
clearly, quickly. Partial Contents: Forms, Generalization, Classes, Deductive System of 
Classes, Algebra of Logic, Assumptions of Principia Mathematica, Logistics, Proofs of 
Theorems, etc. "Clearest . . . simplest introduction .'. . the intelligent non-mathematician 
should have no difficulty," MATHEMATICS GAZETTE. Revised, expanded 2nd edition. Truth- 
value tables. 368pp. 5% 8. S164 Paperbound $1.75 

13 



CATALOGUE OF 

TRIGONOMETRICAL SERIES, Anton i Zygmund. On modern advanced level. Contains carefully 
organized analyses of trigonometric, orthogonal, Fourier systems of functions, with clear 
adequate descriptions of summability of Fourier series, proximation theory, conjugate series, 
convergence, divergence of Fourier series. Especially valuable for Russian, Eastern Euro- 
pean coverage. 329pp. 5% x 8. S290 Paperbound $1.50 

THE LAWS OF THOUGHT, George Boole. This book founded symbolic logic some 100 years 
ago. It is the 1st significant attempt to apply logic to all aspects of human endeavour. 
Partial contents: derivation of laws, signs and laws, interpretations, eliminations, condi- 
tions of a perfect method, analysis, Aristotelian logic, probability, and similar topics, 
xvii + 424pp. 5% x 8. S28 Paperbound $2.00 

SYMBOLIC LOGIC, C. I. Lewis, C. H. Langford. 2nd revised edition of probably most cited 
book in symbolic logic. Wide coverage of entire field; one of fullest treatments of paradoxes; 
plus much material not available elsewhere. Basic to volume is distinction between logic 
of extensions and intensions. Considerable emphasis on converse substitution, while matrix 
system presents supposition of variety of non-Aristotelian logics. Especially valuable sec- 
tions on strict limitations, existence theorems. Partial contents: Boole-Schroeder algebra- 
truth value systems, the matrix method; implication and deductibility; general theory of 
propositions; etc. "Most valuable," Times, London. 506pp. 5% x 8. S170 Paperbound $2.00 



GROUP THEORY AND LINEAR ALGEBRA, SETS, ETC. 

LECTURES ON THE ICOSAHEDRON AND THE SOLUTION OF EQUATIONS OF THE FIFTH DEGREE, 
Felix Klein. Solution of quintics in terms of rotations of regular icosahedron around its 
axes of symmetry. A classic, indispensable source for those interested in higher algebra, 
geometry, crystallography. Considerable explanatory material included. 230 footnotes, mostly 
bibliography. "Classical monograph . . . detailed, readable book," Math. Gazette. 2nd edi- 
tion, xvi + 289pp. 5% x 8. S314 Paperbound $1.85 

INTRODUCTION TO THE THEORY OF GROUPS OF FINITE ORDER, R. Carmichael. Examines 
fundamental theorems and their applications. Beginning with sets, systems, permutations, 
etc., progresses in easy stages through important types of groups: Abelian, prime power, 
permutation, etc. Except 1 chapter where matrices are desirable, no higher math is needed. 
783 exercises, problems, xvi + 447pp. 5% x 8. S299 Clothbound $3.95 

S300 Paperbound $2.00 

THEORY OF GROUPS OF FINITE ORDER, W. Burnside. First published some 40 years ago, 
still one of clearest introductions. Partial contents: permutations, groups independent of 
representation, composition series of a group, isomorphism of a group with itself, Abelian 
groups, prime power groups, permutation groups, invariants of groups of linear substitu- 
tion, graphical representation, etc. "Clear and detailed discussion . . . numerous problems 
which are instructive," Design News, xxiv + 512pp. 5% x 8. S38 Paperbound $2.45 

COMPUTATIONAL METHODS OF LINEAR ALGEBRA, V. N. Faddeeva, translated by C. D. Benster. 
1st English translation of unique, valuable work, only one in English presenting systematic 
exposition of most important methods of linear algebra — classical, contemporary. Details 
of deriving numerical solutions of problems in mathematical physics. Theory and practice. 
Includes survey of necessary background, most important methods of solution, 'for exact, 
iterative groups. One of most valuable features is 23 tables, triple checked for accuracy, 
unavailable elsewhere. Translator's note, x + 252pp. 5% x 8. S424 Paperbound $1.95 

THE CONTINUUM AND OTHER TYPES OF SERIAL ORDER, E. V. Huntington. This famous book 
gives a systematic elementary account of the modern theory of the continuum as a type 
of serial order. Based on the Cantor-Dedekind ordinal theory, which requires no technical 
knowledge of higher mathematics, it offers an easily followed analysis of ordered classes, 
discrete and dense series, continuous series, Cantor's trahsfinite numbers. "Admirable 
introduction to the rigorous theory of the continuum . . . reading easy," Science Progress. 
2nd edition, viii + 82pp. 5% x 8. S129 Clothbound $2.75 

S130 Paperbound $1.00 

THEORY OF SETS, E. Kamke. Clearest, amplest introduction in English, well suited for inde- 
pendent study. Subdivisions of main theory, such as theory of sets of points, are discussed, 
but emphasis is on general theory. Partial contents: rudiments of set theory, arbitrary sets, 
their cardinal numbers, ordered sets, their order types, well-ordered sets, their cardinal 
numbers, vii 4- 144pp. 5% x 8. S141 Paperbound $1.35 

CONTRIBUTIONS TO THE FOUNDING OF THE THEORY OF TRANSFINITE NUMBERS, Georg Cantor. 

These papers founded a new branch of mathematics. The famous articles of 1895-7 are 
translated, with an 82-page introduction by P. E. B. Jourdain dealing with Cantor, the 
background of his discoveries, their results, future possibilities, ix + 211pp. 5% x 8. 

S45 Paperbound $1.25 

14 



DOVER SCIENCE BOOKS 

NUMERICAL AND GRAPHICAL METHODS, TABLES 

JACOB IAN ELLIPTIC FUNCTION TABLES, L. M. Milne-Thomson. Easy-to-follow, practical, not 
only useful numerical tables, but complete elementary sketch of application of elliptic 
functions. Covers description of principle properties; complete elliptic integrals; Fourier 
series, expansions; periods, zeros, poles, residues, formulas for special values of argument; 
cubic, quartic polynomials; pendulum problem; etc. Tables, graphs form body of book: 
Graph, 5 figure table of elliptic function sn (u m); en (u m); dn (u m). 8 figure table of 
complete elliptic integrals K, K', E, E', nome q. 7 figure table of Jacobian zeta-function 
Z(u). 3 figures, xi + 123pp. 5% x 8. S194 Paperbound $1.35 

TABLES OF FUNCTIONS WITH FORMULAE AND CURVES, E. Jahnke, F. Emde. Most comprehensive 
1-volume English text collection of tables, formulae, curves of transcendent functions. 4th 
corrected edition, new 76-page section giving tables, formulae for elementary functions not 
in other English editions. Partial contents: sine, cosine, logarithmic integral; error integral; 
elliptic integrals; theta functions; Legendre, Bessel, Riemann, Mathieu, hypergeometric 
functions; etc. "Out-of-the-way functions for which we know no other source." Scientific 
Computing Service, Ltd. 212 figures. 400pp. ; 5% x 8%. S133 Paperbound $2.00 

MATHEMATICAL TABLES, H. B. Dwight. Covers in one volume almost every function of im- 
portance in applied mathematics, engineering, physical sciences. Three extremely fine 
tables of the three trig functions, inverses, to 1000th of radian; natural, common logs; 
squares, cubes,- hyperbolic functions, inverses; (a 2 + b 2 ) exp. Via,- complete elliptical in- 
tegrals of 1st, 2nd kind; sine, cosine integrals; exponential integrals; Ei(x) and Ei( — x); 
binomial coefficients; factorials to 250; surface zonal harmonics, first derivatives; Bernoulli, 
Euler numbers, their logs to base of 10; Gamma function; normal probability integral; over 
60pp. Bessel functions; Riemann zeta function. Each table with formulae generally used, 
sources of more extensive tables, interpolation data, etc. Over half have columns of 
differences, to facilitate interpolation, viii + 231pp. 5% x 8. S445 Paperbound $1.75 

PRACTICAL ANALYSIS, GRAPHICAL AND NUMERICAL METHODS, F. A. Willers. Immensely prac- 
tical hand-book for engineers. How to interpolate, use various methods of numerical differ- 
entiation and integration, determine roots of a single algebraic equation, system of linear 
equations, use empirical formulas, integrate differential equations, etc. Hundreds of short- 
cuts for arriving at numerical solutions. Special section on American calculating machines, 
by T. W. Simpson. Translation by R. T. Beyer. 132 illustrations. 422pp. 5% x 8. 

S273 Paperbound $2.00 

NUMERICAL SOLUTIONS OF DIFFERENTIAL EQUATIONS, H. Levy, E. A. Baggott. Comprehensive 
collection of methods for solving ordinary differential equations of first and higher order 
2 requirements: practical, easy to grasp; more rapid than school methods. Partial contents: 
graphical integration of differential equations, graphical methods for detailed solution. 
Numerical solution. Simultaneous equations and equations of 2nd and higher orders. 
"Should be in the hands of all in research and applied mathematics, teaching," Nature. 
21 figures, viii + 238pp. 5% x 8. S168 Paperbound $1.75 

NUMERICAL INTEGRATION OF DIFFERENTIAL EQUATIONS, Bennet, Milne, Bateman. Unabridged 
republication of original prepared for National Research Council. New methods of integration 
by 3 leading mathematicians: "The Interpolational Polynomial," "Successive Approximation," 
A. A. Bennett, "Step-by-step Methods of Integration," W. W. Milne. "Methods for Partial 
Differential Equations," H. Bateman. Methods for partial differential equations, solution 
of differential equations to non-integral values of a parameter will interest mathematicians, 
physicists. 288 footnotes, mostly bibliographical. 235 item classified bibliography. 108pp. 
5% X 8. S305 Paperbound $1.35 



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(continued from inside front cover} 



Dynamics of a System of Rigid Bodies (Advanced Section), E. J. Routh 

$235 

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An Elementary Survey of Celestial Mechanics, Y. Ryabov $1.25 

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THE BINARY STARS 



BY ROBERT G. AITKEN 



Originally published in 1918 and completely revised in 1935, this book is still a 
"bible" in the field of double-star astronomy. Its author, a director of the famous 
Lick Observatory, is widely recognized as the father of the modern study of binary 
star systems and, in this book, he sums up the results of centuries of research, 
including 40 years of his own work, and surveys the methods of observation and of 
orbital computation that have been developed. 

The book begins with an extensive two-part historical sketch that covers various 
steps in the discovery and interpretation of binary stars and assesses the work of 
many of the major contributors. There follow chapters on observational methods for 
visual binary stars; the orbit of a visual binary star; the radial velocity of a star 
(written by Dr. J. K Moore of the University of California); the orbit of a spectro- 
scopic binary star; eclfpsing binary stars; the known orbits of the binary stars; some 
binary stars of special interest: Alpha Centauri, Sirius, Krueger 60, 61 Cygni, 
Delta Equulei, Capella, Algol, and others; and the origin of binary stars. Within each 
of these topics, the author emphasizes method, telling how to plot orbits, where to 
locate important data and how to use them in computations, how to measure and 
reduce spectrograms, when to use eyepieces in observations, and so on. This practical 
material is truly unique, and it should prove especially useful in classroom instruc- 
tion, to astronomy students, and to advanced hobbyists. 

Each chapter ends with a bibliography and there is additional, up-to-date bibli- 
ographical material in the notes added to this new edition by Professor J. T. Kent. 
Professor Kent has also revised and corrected the text. Now, more than ever, this 
book is an unequalled source of information on methods of study and as a thorough 
summary of essential binary star material. 

Revised (1935) edition, corrected and with additional notes new to 1963 edition by 
Prof. J. T. Kent. New preface. 50 tables; 13 figures; 4 full-page plates. Bibliographies. 
Appendix: Table of Orbits of Visual and Spectroscopic Binary Stars. Indexes, x + 
309pp. 5% x m. S1102 Paperbound $2.00 



A DOVER EDITION DESIGNED FOR YEARS OF USE! 

We have spared no pains to make this the best book possible. Our paper is opaque, 
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held together with glue. The binding will not crack and split. This is a permanent 
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